STRENGTH, ELASTICITY AND PHASE TRANSITION STUDY ON NACL AND
MGO-NACL MIXTURE TO MANTLE PRESSURES
(Spine title: Strength, Elasticity and Phase Transition Study on NaCl and MgO-NaCl
Mixture to Mantle Pressures)
(Thesis format: Monograph)
by
Zhongying Mi
Graduate Program in Geophysics
A thesis submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
The School of Graduate and Postdoctoral Studies
The University of Western Ontario
London, Ontario, Canada
© Zhongying Mi 2013
THE UNIVERSITY OF WESTERN ONTARIO
School of Graduate and Postdoctoral Studies
CERTIFICATE OF EXAMINATION
Supervisor
Examiners
______________________________
Dr. Sean Shieh
______________________________
Dr. Dazhi Jiang
Supervisory Committee
______________________________
Dr. Rick Secco
______________________________
______________________________
Dr. Mahi Singh
______________________________
______________________________
Dr. John Tse
The thesis by
Zhongying MI
entitled:
Strength, Elasticity and Phase Transition Study on NaCl and Its Mixture to Mantle
Pressures
is accepted in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
______________________
Date
_______________________________
Chair of the Thesis Examination Board
ii
Abstract
Rheological properties of the Earth control most of the important geological
processes, such as mantle convection, plate tectonics, earthquakes and nature of thermal
evolution. Most parts of the Earth consist of multi-phase polycrystalline aggregates with
various composition and compressibility. Therefore, deformation studies on multi-phase
materials are important to understand the rheological properties and convection of the Earth.
NaCl and MgO with large contrast in elastic properties are excellent analogue materials for
modelling the Earth that is generally made of both strong and weak materials. In addition,
NaCl and MgO are widely used as pressure transmitting medium and pressure calibration
standard for high pressures research. Therefore, study of NaCl and MgO at high pressure
should provide meaningful information of stress, strain and strength for mantle dynamics and
also help to construct a two-phase deformation model.
In this study, four different starting materials, pure NaCl, MgO-NaCl (1:3), MgONaCl (1:2), MgO-NaCl (1:1) mixture, were investigated in situ at mantle pressures using
radial X-ray diffraction technique at beamline X17C, National Synchrotron Light Source
(NSLS), Brookhaven National Laboratory (BNL) and pressure gradient method at
Department of Earth Sciences, Western University. Pure NaCl was studied by radial energy
dispersive X-ray diffraction (EDXD) at a pressure up to 43.7 GPa. MgO-NaCl (1:3), MgONaCl (1:2) and MgO-NaCl (1:1) were investigated at pressure to 57.6, 43.1 and 44.4 GPa
respectively by radial angular dispersive X-ray diffraction (ADXD). Pressure gradient
method was only applied to pure NaCl to 55.8 GPa.
It is well known that NaCl undergoes a phase transition from B1 (rock-salt structure)
to B2 (CsCl structure) under high pressure. In this study, the B1 to B2 phase transition in the
mixed systems varies with pressures, including both initial and completion pressures. The
experimental results show that the higher the volume ratio of MgO, the higher pressure is
needed for NaCl to start and complete the phase transition. Therefore, the involvement of
strong materials (MgO) extends the stability of NaCl B1 phase to higher pressures. The
differential stresses (lower bound of yield strength) of NaCl are also varied with different
starting materials. The highest differential stresses occurred in the mixture with highest
iii
volume ratio of “strong” material MgO. It therefore can be inferred that in a mixture, strong
material could strengthen the “soft” material. Conversely, the stresses supported by MgO in
the MgO-NaCl mixture are lower than that of pure MgO, indicating that “soft” material may
also has influence on strong material.
Generally, the strength of NaCl B1 phase increase gently with pressure, suggesting
that NaCl B1 phase is a good pressure transmitting medium, whereas B2 is no longer
regarded as “soft” material due to the abruptly increment of its differential stress. Results
from peak broadening study show that deformation of NaCl B1 phase remains in elastic
regime, whereas B2 phase undergoes a plastic deformation. The elastic constants of B1 phase
calculated based on lattice strain theory show reasonable agreement with previous reports
within B1 phase regime, whereas the elastic constants of B2 phase appear to deviate largely
from the theoretical predictions. Differential stresses supported by different crystal planes
show that B1 200 has the lowest value, suggesting B1 200 may be responsible for the
pressure induced initial deformation.
Keywords
DAC, NaCl, MgO-NaCl mixture, differential stress, strain, phase transformation, synchrotron
X-ray, high pressure
iv
Acknowledgments
Firstly, I would like to thank my supervisor, Dr. Sean Shieh, for his helpful advices,
constructive discussions, patient instructions and support for the past four years.
Secondly, I would like to thank Zhiqiang Chen for his patient, responsible work with
synchrotron facility in Brookhaven National Laboratory. His kindly technical assistance
made my experiments went smoothly. Thanks to Susannah McGregor Dorfman Wong, with
her help I loaded my first small gold piece in synchrotron.
Thanks to my colleagues Sanda Botis, Weiguang Shi and Linada Kaci. My research life has
more fun with their fruitful input.
I would like to thank my friends Jing Chao, Zhaohui Dong, Zhenzhen Huang and Yanyu
Xiao. They color my life and amazingly show how to balance family and research, be both
happy in family life and science.
Most importantly, with full of my heart, I’d like to thank to my family for their support and
encouragement. Without their help and understanding I would not have been able to finish
this work.
v
Table of Contents
CERTIFICATE OF EXAMINATION ........................................................................... ii
Abstract .............................................................................................................................. iii
Acknowledgments............................................................................................................... v
Table of Contents ............................................................................................................... vi
List of Tables ...................................................................................................................... x
List of Figures .................................................................................................................... xi
List of Appendices ........................................................................................................... xix
List of Abbreviations ........................................................................................................ xx
Chapter 1 ............................................................................................................................. 1
1 Introduction .................................................................................................................... 1
1.1 Rheology of the Earth ............................................................................................. 1
1.2 Multi-phase Structure of the Earth .......................................................................... 3
1.3 Deformation of Multi-phase Materials ................................................................... 4
1.4 Experimental Techniques to Study Deformation .................................................... 7
1.5 NaCl, Mixture of NaCl and MgO ........................................................................... 9
1.5.1
Sodium Chloride ......................................................................................... 9
1.5.2
Magnesium Oxide ..................................................................................... 10
1.5.3
Mixture of NaCl and MgO ........................................................................ 11
1.6 Objectives and structure of this thesis .................................................................. 12
Chapter 2 ........................................................................................................................... 15
2 Experiments and Instruments ....................................................................................... 15
2.1 The Diamond Anvil Cell ....................................................................................... 15
2.2 Synchrotron X-ray Radiation ................................................................................ 19
2.3 Powder X-ray Diffraction ..................................................................................... 23
vi
2.4 Raman Spectroscopy and Ruby Fluorescence ...................................................... 25
2.4.1
Raman Spectroscopy................................................................................. 25
2.4.2
Ruby Fluorescence Scale .......................................................................... 27
2.5 Sample Preparation ............................................................................................... 28
Chapter 3 ........................................................................................................................... 30
3 Theories ........................................................................................................................ 30
3.1 The Lattice Strain Theory ..................................................................................... 30
3.2 Pressure Gradient Method..................................................................................... 35
3.3 Peak Width Theory ............................................................................................... 37
Chapter 4 ........................................................................................................................... 40
4 Strength and Elasticity Study on NaCl......................................................................... 40
4.1 Experiment ............................................................................................................ 41
4.2 Results and Discussions ........................................................................................ 42
4.2.1
X-ray Diffraction Patterns......................................................................... 42
4.2.2
Phase Transformation ............................................................................... 44
4.2.3
Strength Study........................................................................................... 45
4.2.4
Strain Study............................................................................................... 51
4.2.5
Elasticity Study ......................................................................................... 52
4.3 Conclusion ............................................................................................................ 53
Chapter 5 ........................................................................................................................... 55
5 Strength and Elasticity Study on MgO-NaCl (1:3) Mixture ........................................ 55
5.1 Experiment ............................................................................................................ 55
5.2 Results and Discussions ........................................................................................ 57
5.2.1
2-D Images ................................................................................................ 57
5.2.2
1-D Diffraction Patterns ............................................................................ 59
5.2.3
Phase Transformation ............................................................................... 61
vii
5.2.4
Strength Study........................................................................................... 63
5.2.5
Elasticity Study ......................................................................................... 68
5.3 Conclusions ........................................................................................................... 70
Chapter 6 ........................................................................................................................... 71
6 Strength and Elasticity Study on MgO-NaCl (1:2) Mixture ........................................ 71
6.1 Experiment ............................................................................................................ 71
6.2 Results and Discussions ........................................................................................ 72
6.2.1
1-D Diffraction Patterns ............................................................................ 72
6.2.2
Phase Transformation ............................................................................... 74
6.2.3
Strength of NaCl ....................................................................................... 76
6.2.4
Elasticity Study of NaCl ........................................................................... 80
6.2.5
Strain of NaCl ........................................................................................... 81
6.2.6
Strain of MgO ........................................................................................... 83
6.3 Conclusions ........................................................................................................... 84
Chapter 7 ........................................................................................................................... 86
7 Strength and Strain Study on MgO-NaCl (1:1) Mixture.............................................. 86
7.1.1
Experiments .............................................................................................. 86
7.2 Results and Discussions ........................................................................................ 87
7.2.1
1-D Diffraction Patterns ............................................................................ 87
7.2.2
Phase Transformation ............................................................................... 89
7.2.3
Strength of NaCl ....................................................................................... 93
7.2.4
Strain of NaCl ........................................................................................... 97
7.2.5
Strength and Strain Study of MgO............................................................ 98
7.3 Conclusions ......................................................................................................... 103
Chapter 8 ......................................................................................................................... 105
8 Conclusions ................................................................................................................ 105
viii
References ....................................................................................................................... 108
Appendices ...................................................................................................................... 121
Curriculum Vitae ............................................................................................................ 126
ix
List of Tables
Table 1-1 Summary staring materials, experimental methods, and pressure ranges for the
NaCl and MgO-NaCl mixture study in this thesis .................................................................. 14
Table 5-1. Comparisons of NaCl B1 and B2 phase transition in different studies. ................ 63
Table 6-1 Phase boundary between B1 and B2 phase with different vol% of MgO .............. 75
Table 7-1 Comparison of NaCl B1 and B2 phase transition in different studies ................... 91
x
List of Figures
Figure 1.1. Mineralogical model of the Earth’s crust and mantle (Bovolo, 2005). The vertical
axis denotes the depth and the horizontal axis indicates the volume ratio of different minerals
in the deep Earth. Ca-pv: calcium perovskite; px: pyroxene. ................................................... 4
Figure 1.2. Schematic plot of the stress-strain behavior of material. The material exhibits
elastic behavior until sufficient stress is applied to reach the yield strength, at which point
permanent deformation occurs. In the elastic range, the stress/strain ratio determines the
elastic modulus.......................................................................................................................... 5
Figure 1.3. Crystal structure of NaCl B1 and B2 phases. (a) B1 structure; (b) B2 phase. Green
balls are Na atoms, blue balls are Cl atoms. B1 phase has a rock-salt structure with 6
coordination numbers. B2 phase has a CsCl structure with 8 coordination numbers............. 10
Figure 2.1. Pressures achieved by modern static compression methods and the corresponding
pressures at depth in the Earth (Geotherm), DAC-diamond anvil cell; LVP-large volume
press; RHDAC-resistive heating diamond anvil cell; LHDAC-laser heating diamond anvil
cell; together with the approximate P-T ranges for planets. ................................................... 16
Figure 2.2. Schematic of diamond anvil cell. (a) compress-screws pushing diamonds against
each other to produce pressure; (b) indicates sample chamber............................................... 17
Figure 2.3. Types of diamond cutting. (a) Brilliant (b) Drukker types ................................... 18
Figure 2.4 A beryllium gasket in between two diamonds tips ................................................ 19
Figure 2.5. Schematic view of a synchrotron radiation source. The injection-accelerating
system (Linac+Booster synchrotron) is inside the storage ring which is actually like a
polygon with the bending magnets at their vertices. Bending magnets, BM, provoke the
deflection of the electron trajectory and as a consequence produce the synchrotron radiation
which escapes forwardly. Insertion devices, ID, can be positioned in the straight sectors to
produce specific synchrotron light. The radiofrequency cavity of the storage ring and those of
the booster synchrotron are also indicated. ............................................................................. 20
xi
Figure 2.6. A historical graph shows the brilliance of X-ray sources from bending magnet
(BM), wiggler and undulator (Willmott, 2011). ..................................................................... 21
Figure 2.7. Comparative illustration of the generation process of synchrotron light by a
bending magnet (a) and by two different insertion devices (flat undulator b and wiggler c):
periodic, weak deflection of the electron beam in a flat undulator of period length λp (λ:
wavelength; p: period) and larger transverse oscillations of the beam produced in a wiggler.
Differences in photon beam collimation achieved for the three magnetic elements are shown
................................................................................................................................................. 22
Figure 2.8. Schematic angle dispersive X-ray diffraction (ADXD) setup using a DAC at a
radial geometry ....................................................................................................................... 24
Figure 2.9. Schematic energy dispersive X-ray diffraction (EDXD) setup using DAC at a
radial geometry ....................................................................................................................... 25
Figure 2.10. Layout of a Raman system used in this study for high pressure ruby fluorescence
measurements. ......................................................................................................................... 27
Figure 2.11. Ruby R-line spectra at pressures of 10 and 18.79 GPa. R-lines shift to the high
frequency direction as pressure increase................................................................................. 28
Figure 2.12. Example of sample configuration for the radial X-ray diffraction study in a DAC.
NaCl sample is shown by white color and a small piece of yellow shiny gold foil lies in the
center of beryllium sample chamber. ...................................................................................... 29
Figure 3.1. Schematic illustration of diffraction geometry in a radial X-ray experiment in the
diamond cell, σ3 is the axial stress while σ1 is the radial stress, ψ is the diffraction angle. ... 31
Figure 3.2. Schematic plot of stress distribution inside a diamond anvil cell (modified from
Meade and Jeanloz, 1988). The ruby spheres (red dots) are distributed across the sample
chamber and are used for pressure determination................................................................... 36
Figure 4.1. EDXD experiment set up in a radial geometry at beamline X17C, NSLS,
Brookhaven National Laboratory. .......................................................................................... 42
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Figure 4.2. Representative X-ray diffraction patterns of NaCl B1 phase obtained at different
ψ angles at 29.8 GPa. B1: NaCl B1 phase; B2: NaCl B2 phase; Au: gold; Be: beryllium
gasket. The ψ angles were labeled with each pattern. The arrows indicate the shifts of the
peak under different stress condition. ..................................................................................... 43
Figure 4.3. Diffraction patterns of NaCl collected at hydrostatic condition up to 43.7 GPa.
The arrows show the emergence of NaCl B2 phase started at 29.8 GPa and accomplished
fully at 32.3 GPa. B1 denotes NaCl B1 phase, B2 denotes NaCl B2 phase, Au denotes gold
and Be denotes beryllium. Pressure is labeled with each pattern at the right hand side of the
Figure. ..................................................................................................................................... 44
Figure 4.4. d-spacings as a function of 1-3cos2ψ for NaCl, B1 111 200 and and B2 110 and
200. The pressure is labeled next to each fitted lines. ............................................................ 46
Figure 4.5. The ratio of differential stress to shear modulus as a function of pressure for B1
and B2 phase of NaCl. ............................................................................................................ 47
Figure 4.6. Ratios of differential stress to shear modulus value of B1 phase as a function of
pressure for different planes of NaCl. ..................................................................................... 48
Figure 4.7. The differential stress supported by different planes of NaCl as a function of
pressure. Solid symbols and lines are from this study whereas open symbols and dashed lines
are from Funamori et al. (1994) .............................................................................................. 49
Figure 4.8. Comparison of differential stress supported by NaCl at different pressures
obtained from this study with previous published reports (Kinsland and Bassett, 1977; Meade
and Jeanloz, 1988; Weidner et al., 1994; Funamori et al., 1994). Solid green symbols and
green line are NaCl data from RXRD, whereas solid pink symbols are NaCl data from
pressure gradient method. ....................................................................................................... 51
Figure 4.9. a) Strain as a function of stress for NaCl B1 phase; b) strain as a function of
pressure for NaCl B1 and B2 phases. The green symbols and lines represent NaCl B1 phase
and purple symbols stand for B2 phase. ................................................................................. 52
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Figure 4.10. Comparison of single crystal elastic constants of NaCl obtained from the lattice
strain theory and theoretical calculations. Solid circles represent B1 phase, solid diamonds
indicate B2 phase. Open symbols are from previous studies (Xiao et al., 2006; Bass et al.,
2006; Whitefield et al., 1976). ................................................................................................ 53
Figure 5.1. ADXD experiment set up in a radial geometry at X17C, NSLS. ......................... 56
Figure 5.2. (a) A typical 2-D image of CeO2 recorded in this work; (b) a typical caked
diffraction image deduced from 2-D rings.............................................................................. 57
Figure 5.3. 2-D image collected at 5.26, 31.7 and 57.6 GPa respectively. The corresponding
caked images are shown at certain two theta angles to emphasize the development of the
stress upon the sample. ........................................................................................................... 58
Figure 5.4. 1-D diffraction patterns collected at 5.26, 31.7 and 57.6 GPa respectively. B1:
NaCl B1, B2: NaCl B2, Au: gold, Be: beryllium. .................................................................. 61
Figure 5.5. Diffraction patterns of MgO-NaCl (1:3) mixture collected at hydrostatic condition
up to 57.6 GPa. Arrows show B1 to B2 phase transition started at 29.4 GPa and
accomplished at 36.7 GPa. ...................................................................................................... 62
Figure 5.6. Plot of d-spacing versus ψ angle shows sinusoid curves for NaCl B1 111 and B2
110. The corresponding d-spacing as a function of 1-3cos2 ψ shows linear fits for both
phases. Pressure values are labeled on the right vertical axis. ................................................ 65
Figure 5.7. Plot of differential stress over shear modulus as a function of pressure for MgONaCl (1:3) mixture. Miller indices are labeled with each line. It shows that the increment of
differential stress of B1 111 over shear modulus is the greater than other planes of B1 phase.
................................................................................................................................................. 66
Figure 5.8. Differential stress supported by NaCl at different planes at pressure up to 57.6
GPa. NaCl B1 phase shows a lower differential stress than those of B2 phase, indicating a
better pressure medium than B2. ............................................................................................ 67
Figure 5.9. Comparison of differential stress supported by NaCl B1 and B2 phases at
pressure to 56 GPa. Solid symbols are from this study and open symbols are from previous
xiv
studies (Kinsland and Bassett, 1977; Meade and Jeanloz, 1988; Funamori et al., 1994;
Weidner et al., 1994) .............................................................................................................. 68
Figure 5.10. Comparisons of single crystal elastic constants of NaCl obtained from the lattice
strain theory (NaCl in mixture), theoretical calculations and Brillouin data (pure NaCl). Solid
symbols are from this study. Open symbols and dashed and dash-dotted lines are from
previous published reports (Xiao et al., 2006; Bass et al., 2006; Whitefield et al., 1976). .... 69
Figure 6.1. (a) Diffraction patterns of MgO-NaCl (1:2) were collected at 43.1 GPa. Miller
indices are labeled with peaks. B2 denotes NaCl B2 phase, Be indicates beryllium, Au
represents gold. The ψ angles from 0° to 180° are labeled with peaks. (b) The zoom-in plot
with two theta at 10.5-12.5 shows that smallest two theta value appears at ψ=90° and biggest
values at both 0° and 180°. Dashed line shows two theta changing tendency. ...................... 73
Figure 6.2. Diffraction patterns of MgO-NaCl (1:2) mixture collected at hydrostatic condition
up to 43.1 GPa. The arrows show the emergence of B2 phase started at 30.7 GPa and
accomplished fully at 38.8 GPa. Pressures are labeled at the right vertical axis. ................... 74
Figure 6.3. Plot of NaCl B2 110 d-spacing versus ψ angle shows sinusoid curves from 33.5 to
43.1 GPa. Pressure (hydrostatic pressure) is labeled with the curves. The d-spacings are
decreasing with pressure. The largest d-spacings are at around 90° for all pressures. ........... 76
Figure 6.4. NaCl B2 110 d-spacings as a function of 1-3cos2ψ at pressure of 33.5-43.1 GPa.
Linear fits can be clearly observed. Pressure is labeled with each linear fit. The d-spacings
decrease as pressures increase. Dashed line indicates the intercept and also the hydrostatic
condition. ................................................................................................................................ 77
Figure 6.5. Plot of differential stresses over shear modulus as function of pressures for NaCl
in MgO-NaCl (1:2) mixture. Miller indices are labeled with each line. It shows that the
increment of differential stress of B1 111 over shear modulus is greater than other planes of
B1 phase. For B2 phase, the B2 200 shows the highest value than other planes. .................. 78
Figure 6.6. Differential stress of NaCl B1 and B2 along different planes at high pressures.
Miller indices are labeled with each plane. NaCl B1 200 has the lowest differential stress
(yield strength), which means B1 200 may yield first under high pressure. .......................... 79
xv
Figure 6.7. Comparison of differential stress of NaCl B1 and B2 phases at elevated pressures.
Solid blue symbols are results from this experiment, MgO-NaCl (1:2) mixture; Solid pink
symbols (MgO-NaCl (1:3) are from chapter 5; Solid green circles (pure NaCl) are from
chapter 4. It clearly shows that the strength of NaCl (with 33.3% MgO) is larger than that of
NaCl (with 25% MgO) and pure NaCl, indicating the strong material enhances the strength
of the soft material. Open symbols are from previous studies (Kinsland and Bassett, 1977;
Meade and Jeanloz, 1988; Funamori et al., 1994; Weidner et al., 1994) ............................... 80
Figure 6.8. Comparison of single crystal elastic constants of NaCl obtained from the lattice
strain theory, theoretical calculations and Brillion data. Solid symbols are from this study.
Open symbols, dashed and dash-dotted lines are from previous published reports (Xiao et al.,
2006; Bass et al., 2006; Whitefield et al., 1976) .................................................................... 81
Figure 6.9. Plot of sin2θ versus B2cos2θ variation for NaCl B1 and B2 phases at different
pressures. B: full width at half maximum. Miller indices are labeled with data. The solid lines
are linear fits to the experimental data. The slope of the straight line reflects the grain size.
Within B1 phase range, the slope increases and arrives to highest value at 25.5 GPa; For B2
phase, the strain (slope) increase from 33.5 to 40.6 GPa, while a drop occurred at 43.1 GPa.
................................................................................................................................................. 82
Figure 6.10. Variations of strain as a function of stress for NaCl B1 and B2 phases. Green
symbols and line are NaCl B1 phase, and purple symbols and line denote NaCl B2 phase.
Pressures are labeled with data point. In B1 phase, strain increases linearly with stress, which
suggests an elastic deformation process. For B2 phase, strain increase non-linearly with stress
from 33.5 to 43.1 GPa suggesting a plastic deformation already developed. ......................... 83
Figure 6.11. (a) Plot of sin2θ as a function of B2cos2θ for MgO at different pressure. The
FWHM and two theta values of MgO 111, 200 and 220 peaks were used for the fittings. (b)
The variations of strain as a function of pressure for MgO up to 25.5 GPa. The strain
increases linearly with pressure, which suggests an elastic deformation process. ................. 84
Figure 7.1. Diffraction patterns of MgO-NaCl mixture (1:1) collected at 6.8 GPa were
stacking together in a 5-degree angle step. (a) Miller indices are labeled with peaks. B1
denotes NaCl B1 phase, BeO denotes beryllium Oxide, Au denotes gold. ψ angle from 0° to
xvi
180° is labeled with axis . (b) The zoom-in plot with two theta at 10.5-12.50 shows that
smallest two theta value appears at ψ=90° and largest values at both 0° and 180°. Dashed line
shows two theta changing tendency. ....................................................................................... 88
Figure 7.2. Diffraction patterns of MgO-NaCl (1:1) mixture collected at hydrostatic condition
up to 31.6 GPa. B1 denotes NaCl B1 phase and Au denotes gold. Pressures are labeled with
each pattern. ............................................................................................................................ 89
Figure 7.3 Diffraction patterns of MgO-NaCl (1:1) mixture collected at hydrostatic condition
up to 44.4 GPa. The arrows show the emergence of NaCl B2 phase started at 33.1 GPa and
accomplished fully at 38.8 GPa. B1 denotes NaCl B1 phase, B2 denotes NaCl B2 phase, Au
denotes gold and Be denotes beryllium. Pressure is labeled with each pattern. ..................... 90
Figure 7.4 B1-B2 transition pressure for MgO-NaCl binary system. Solid green symbols
represent the emergence of B2 phase and solid pink circles represent the completion of B1B2 phase transition. ................................................................................................................. 92
Figure 7.5 Plot of NaCl B1 110 and MgO 200 d-spacings versus ψ angles show sinusoid
curves from 6.7 to 23.6 GPa. Pressure (hydrostatic condition) is labeled with each curve. The
d-spacings are decreasing with pressure. The largest d-spacing shows at around 90° for all
pressures. ................................................................................................................................. 93
Figure 7.6 NaCl B1 111 and MgO 200 d-spacings as a function of 1-3cos2 ψ at pressures of
6.7-23.6 GPa. Linear fits can be clearly observed. Pressure is labeled with each linear fit. The
d-spacings decrease as pressures increase. Dashed lines indicate the intercept and also the
hydrostatic condition. .............................................................................................................. 94
Figure 7.7 Plots of differential stresses over shear modulus as a function of pressures for
MgO-NaCl (1:1) mixture. B1 denotes NaCl B1 phase and B2 denotes NaCl B2 phase. B1
111 shows the highest values for B1 phase. For B2 phase B2 200 shows the highest value. 95
Figure 7.8 Comparison of differential stresses of NaCl B1 and B2 phases at elevated
pressures with previous studies (Kinsland and Bassett, 1977; Meade and Jeanloz, 1988;
Weidner et al., 1994; Funamori et al., 1994). Solid orange symbols are results from this
experiment, MgO-NaCl (1:1) mixture; Solid blue symbols MgO-NaCl (1:2) are from chapter
xvii
6; Solid pink circles (MgO-NaCl (1:3)) are from chapter 5; Solid green circles (pure NaCl)
are from chapter 4. It clearly shows that the strength of NaCl (with 50% MgO) is larger than
that of NaCl (with 33.3% MgO), NaCl (with 25% MgO) and pure NaCl, indicating the strong
material enhances the strength of the soft material. ................................................................ 96
Figure 7.9 Plot of sin2θ versus B2cos2θ variation for NaCl B2 phases at different pressures.
The strain (slope) increase from 34.8 to 39.5 GPa, while a drop occurs at 42.3 GPa. ........... 97
Figure 7.10 Variations of strain as a function of stress for NaCl in this study. Green circles
and line are NaCl B1 phase whereas pink symbols and line denote NaCl B2 phase. Pressures
are labeled with data point. In B1 phase, strain increases linearly with stress from 39.5 to
44.4 GPa suggesting a plastic deformation already developed............................................... 98
Figure 7.11 Comparison of differential stresses of MgO as a function of pressure obtained
from this study with previous data (Kinsland and Bassett, 1977; Meade and Jeanloz, 1988;
Duffy et al., 1995; Merkel et al., 2002; Uchida et al., 2004; Singh et al., 2004; Akhmetov,
2008; Gleason et al., 2011). Solid red circles: this study; red line: power fit for Singh, 2004;
green open diamonds: run1 and run 2 in Merkel, 2002; green line: linear fit from Duffy, 1995;
black line: linear fit from Meade and Jeanloz, 1988, large strain; blue line: linear fit for
Akhmetov, 2008; purple and black circle: Kinsland and Bassett, 1977; blue dashed lines:
Uchida et al., 2004 ................................................................................................................ 101
Figure 7.12 Plot of sin2θ versus B2cos2θ variation for MgO phases at from 6.7 to 23.6 GPa.
Miller indices are labeled with data. The slopes become steeper with pressure increase. ... 102
Figure 7.13 The variations of strain as a function of stress for MgO in this study. Pressures
are labeled with data point. The strain increases linearly with stress, which suggests an elastic
deformation for MgO at pressure range of 6.7-23.6 GPa. .................................................... 103
xviii
List of Appendices
Appendix A1: The variations of the d-spacings as a function of (1-3cos2 ψ) for the pure
NaCl ……………………….…………………………………………………….…..…121
Appendix A2: The variations of the d-spacings as a function of (1-3cos2 ψ) for the MgONaCl (1:3) mixture ..……………...……………………………………..…………..….122
Appendix A3: The variations of the d-spacings as a function of (1-3cos2 ψ) for the MgONaCl (1:2) mixture………………………………………………………..…………..…123
Appendix A4: The variations of the d-spacings as a function of (1-3cos2 ψ) for the MgONaCl (1:1) mixture………………………………………………………..…………..…124
xix
List of Abbreviations
ADXD: Angle dispersive X-ray diffraction
Au: gold
Be: beryllium
BM: bending magnets
BNL: Brookhaven National Laboratory
BS: beam splitter
CCD: charge-coupled device
DAC: diamond anvil cell
D-DIA: deformation diamond
EDM: electronic discharge machine
EDXD: energy dispersive X-ray diffraction
Fp: ferropericlase
FWHM: full width at half maximum
GPa: Giga-Pascal
ID: Insertion device
IR: Infrared
KB: Kirkpatrick-Baez (a kind of mirror)
LHDAC: laser heating diamond anvil cell
NF: Notch filter
xx
NSLS: National Synchrotron Light Source
PTM: Pressure transmitting medium
Pv: perovskite
RDA: rotation Drickamer apparatus
rf: Radio frequency
RHDAC: resistive heating diamond anvil cell
XRD: X-ray diffraction
xxi
1
Chapter 1
1
Introduction
1.1 Rheology of the Earth
The Greek root of “rheos” means studying the deformation and flow of matter.
Everything flows to some extent if given a long enough time scale, not even to mention
the Earth’s long geological time scale. Rheological properties of the Earth control most of
the important geological processes, such as mantle convection, plate tectonics,
earthquakes and the nature of thermal evolution. However, investigation of the
rheological properties of the Earth is challenging, it involves extreme pressure,
temperature and long geological time. There are two basic approaches to study the
rheology: continuum mechanics approach and solid state physics approach. In the
continuum mechanics approach, the rheological properties of materials are described
phenomenologically, without much reference to the atomic processes which govern the
macroscopic behavior. In the solid state physics (or microphysical) approach, attention is
focused on properties at the atomic level, and on how these affect the rheological
behavior. These two approaches are defined as macro-rheology and micro-rheology
respectively. On the surface or shallow parts of the Earth, deformation properties can be
observed and described phenomenally. However, large parts of the Earth lie underneath
and cannot be accessed directly by observation. Alternatively, much information can be
gained from in-house experiments using Earth materials in a simulated high pressure and
high temperature environment, in a micro-rheology method.
Therefore, rheology of the Earth studied from the solid state physics point of view
is mainly considered and described in this thesis. In some cases, we have evidence that a
rock came from deep Earth, i.e. mantle, based on the chemical composition or crystal
structure. We can infer the temperature and pressure condition at which the rock formed.
However, understanding the history of deformation is very complicated, because it is
hard to know where it happened and under what stress or strain conditions. To address
these issues, it is necessary to understand the deformation mechanism and process of
2
minerals and rocks using experimental approaches by suitable techniques and methods.
However, experimental studies of deformation of minerals and rocks are not
straightforward. The major reason is that deformation properties of minerals and rocks
are time-dependent in many cases. The response of minerals or rocks to external force
depends on the deformation rate. And in many cases, the rate of deformation under
geological conditions is much lower than typical rate of deformation obtained in the
laboratory. Therefore, high accuracy and low stress experiments, which are largely
depend on the technology developments, are of significant importance and need to be
studied considerably.
Over the last decade, the advances in high-pressure experiment keep making
efforts on technology improvement. So far new technologies have led to significant
advances in material deformation property measurements at high pressures and high
temperatures. Among those technologies, diamond anvil cell (DAC) and D-DIA
(deformation DIA), combined with synchrotron radiation, attract considerable attention
and a great number of high pressure experiments are conducted. However, the
deformation D-DIA method (Wang et al., 2003) only allow scientists to study the
deformation behavior of the Earth materials up to 20 GPa, corresponding to a depth of
about 570 km (Nishiyama et al., 2008; Kawazoe et al., 2010). To perform deformation
studies at pressures to the depth of Earth’s deep mantle, the diamond anvil cell combined
with synchrotron X-ray diffraction are regarded as promising tools (Mao et al., 1998;
Miyagi et al., 2010). Many efforts have been made during the last decades, to obtain
quantitative experimental data on rheology under higher pressures by diamond anvil cell
technique. It is known that rheological properties involve stress, strain and time.
Although strain rate (as a function of time) is difficult to be well controlled in the DACs,
information on stress and strain can be derived by DAC experiments up to very high
pressure, e.g. exceeding 400GPa (Ruoff, et al. 1990).
There are several stress and strength reports at high pressures using DAC.
Kinsland and Bassett (1977) studied the strength of MgO and NaCl polycrystals to 25
GPa at room temperature; Meade and Jeanloz (1988) investigated the strength of NaCl
and MgO individually up to ~40 GPa. Mao et al. (1998) reported the strength and
3
elasticity of iron above 200 GPa. Shieh and colleagues (2002, 2004) studied the strength
and elasticity of SiO2 and Ca-perovskite at mantle pressures. Kavner et al. (2003, 2007)
reported the strength and elasticity of ringwoodite and garnet at mantle pressures. The
texture developments in olivine, ringwoodite, magnesiowüstite and silicate perovskite
were reported by Wenk et al. (2004). Miyagi et al. (2010) investigated the slip systems in
MgSiO3 post-perovskite and provided an explanation for the anisotropy of the D” layer.
However, most of the studies focused on a single phase and knowledge of strength and
stress of two-phase mixture at high pressure remains poorly understood. Therefore, the
primary goal of this study is to construct the strength and rheological model of mixed
phase aggregates for future application on the multi-phase compositions of deep Earth.
1.2 Multi-phase Structure of the Earth
Most parts of the Earth consist of multi-phase polycrystalline aggregates with various
compositions (Figure 1.1). The upper mantle mainly consists of olivine, garnet and
pyroxene. This is well known because upper mantle rocks are sometimes brought to
surface either as huge tectonic fragments or as small inclusions in volcanic rocks (like
Kimberlite). Between the upper and lower mantle lie a transition zone extending from
410 to 660 km. High pressure experiments on upper mantle material reveal that the
minerals present there transform to denser phases over a relatively narrow pressure range
corresponding to the pressures and temperatures found in the transition zone and thereby
provide a potential explanation for these discontinuities (Liu, 1979, Ringwood, 1979).
Perovskite and ferropericlase are believed to be the dominant and stable phases in the
lower mantle. The bottom 250-350 km of the lower mantle (known as D” region) is
mainly composed of post-perovskite phase (Murakami et al., 2004; Hirose et al., 2006).
Although constraints on the composition of the lower mantle can be similarly placed, the
exact minerals compositions, their physical properties (e.g. rheological properties of
mixture phases) or quantities of any other minerals or elements that might be present are
still not well constrained yet.
The multi-phase structures of the Earth may result in diverse and complicated
rheological properties. The seismic wave velocities propagated into multi-phase materials
are distinct from a homogeneous single phase, because of the stress-strain distribution as
4
well as the volume fraction and orientation of each phase. In addition, the deformation
mechanism of a multi-phase material is different from a single phase. There are growing
interests to explore how strength of a multi-phase material is related to those of individual
single-phase materials and their volume ratios, geometry and orientation (Li et al., 2007;
Ji et al., 2001; Dresen et al., 1998; Takeda, 1998; Bruhn et al., 1999; Tullis et al., 1991).
Therefore, deformation studies on multi-phase materials are essential and important to
understand the convection and dynamics of the Earth interior.
Figure 1.1. Mineralogical model of the Earth’s crust and mantle (Bovolo, 2005). The
vertical axis denotes the depth and the horizontal axis indicates the volume ratio of
different minerals in the deep Earth. Ca-pv: calcium perovskite; px: pyroxene.
1.3 Deformation of Multi-phase Materials
Deformation can be subdivided into two categories which are elastic and in-elastic,
respectively. When a small stress is applied for a short time, a material will be deformed
instantaneously. The material returns to its initial state when the stress is removed. This
instantaneous and recoverable deformation is called elastic deformation. Elastic constants
can be used to characterize elastic deformation. In contrast, when large enough stress is
5
applied for a long time, the permanent deformation may occur and the material cannot
revert to its original state. This process is called non-elastic deformation. As it is shown
in Figure 1.2, a material is under elastic deformation within elastic region (green line).
Once reached the yield point, then plastic deformation will be initiated (pink curve).
Weidner et al. (2004) suggested that the effectiveness of elastic constants derivation
should be based on material deformed elastically. To ensure the validation of elasticity
study, strain and stress curve become an important way to evaluate the validation of the
elastic constants obtained from this type of study. In this work, deformation experiments
were conducted in DACs, using a radial X-ray diffraction method. All the elastic constant
derivations were based on the assumption that the samples were undergone elastic
deformation.
Figure 1.2. Schematic plot of the stress-strain behavior of material. The material
exhibits elastic behavior until sufficient stress is applied to reach the yield strength,
at which point permanent deformation occurs. In the elastic range, the stress/strain
ratio determines the elastic modulus.
Deformation properties of single-phase materials have been investigated through
laboratory experiments at high pressure and high temperature conditions using DAC, DDIA, multi-anvil cell and shock wave method. However, rheological properties of multi-
6
phase materials are still limited. That is because (1) different phases may encounter a
chemical reaction under high pressure and high temperature conditions; (2) behaviors of
multi-phase aggregate are difficult to interpret. Therefore, choosing proper materials with
no or low chemical reaction potential is critical for a successful high pressure and high
temperature deformation experiments. Knowledge of the deformation properties of
aggregates materials is important for understanding the practical deformation in Earth’s
interior.
It is known that deformation properties involve stress, strain, strength and time.
Yield strength can be described by differential stress (the lower bound of yield strength)
under Von Mises yield criterion. Previous studies on the strength of multi-phase mixture
exhibit a great complexity comparing to its end members.
Hitchings et al. (1989) carried out an experiment on olivine-orthopyroxene
composite and suggested that the strength of mixture was weaker than its end members.
However, McDonnell and colleagues (2000) conducted experiments on olivineorthopyroxene aggregates with orthopyroxene content up to 20 vol %. They concluded
that there was no significant difference of the strength between single phase and mixture.
Ji and colleagues (2001) provided a model which states that the strength of the mixture
would lie somewhere between the end members, based on the forsterite- enstatite
composite experiment. However, strength of a composite could be strengthened with the
lower porosity as suggested by Ford and Wheeler (2004), based on their observations on
a 20% quartz and 80% marble mixture. Heege et al. (2004) reported that the rheological
properties (diffusion creep) depended on the grain sizes. Additionally, Li et al. (2007)
suggested that the strength of the mixture could be varied by different volume ratios of
single phase. In their experiments, spinel (MgAl2O4) and periclase (MgO) were examined
with different volume ratios. They concluded that 40%:60% ratio mixture of MgO and
spinel was the weakest aggregate of all. Therefore, correlation between the strength of
aggregate material and its end members is still not clear.
7
As complex as shown above, more high pressure and high temperature microrheological experiments are needed to clarify strength and rheological properties of the
mixture phases. This is one of the research goals in this study.
1.4 Experimental Techniques to Study Deformation
To quantitatively investigate the rheological properties of minerals and rocks under high
pressures, many efforts have been made to develop appropriate apparatus and advanced
techniques. A great number of deformation experiments have been conducted in worldwide laboratories.
In 1960s, David Griggs developed a high pressure high temperature deformation
device by modifying the piston-cylinder type high-pressure apparatus. Using this
apparatus (the Griggs apparatus), plastic properties and micro-structural development of
rocks were investigated up to ~3 GPa and ~1600K. However, the stress measured by the
Grigg apparatus has large uncertainties, because stress is measured outside of a pressure
vessel in this apparatus and since the effect of friction is very large, sample stress was not
accurately known.
Noticing this issue, efforts over the following ~20 years (1970s-1990s) were
focused on high-accuracy experiments using well-characterized synthetic polycrystalline
aggregates or single crystals. Paterson (1990) designed a gas-medium high-pressure
deformation apparatus. In this device, a deviatoric stress is generated and measured by a
load cell. The friction problem can be avoided because that the load cell is inside the
pressure vessel. High accurate measurements therefore can be achieved since the friction
problem can be overcome. However, the pressure range that can be reached was very
limited (<1 GPa). Such low pressure cannot even give detail information on the
rheological behaviors of the lower continental crust (30-70 km thick, ~2 GPa).
Recently, studies on mechanical properties as well as the deformation
microstructure of the Earth materials are performed by multi-anvil press, rotational
Drickamer apparatus (RDA), deformation-DIA (D-DIA) and diamond anvil cell (DAC).
8
In a multi anvil press, differential stress is generated in the sample space
(Fujimura, et al., 1981; Karato and Rubie, 1997), and the slip systems of wadsleyite and
perovskite were reported (Thurel and Cordier, 2003; Cordier, et al., 2004). The hightemperature and high-pressure environment produced in a multi anvil press is similar to
the Earth’s interior conditions. However, deformation is not steady-state, and most stress
levels are very high and not well understood (Karato and Weidner, 2008).
Torsion experiments can be conducted by RDA because a rotational actuator is
applied and thus one of the anvils can be rotated. Deformation experiments can be
performed at pressures greater than 15 GPa and temperatures beyond 1800K. However,
the disadvantage of this design is that it cannot measure the stress accurately. Due to the
torsion design, stress and strain changes as a function of distance from the center of
rotation.
The D-DIA is modified on the basis of a cubic anvil system from the Japanese
designed DIA. The deformation experiments are conducted at high pressure by moving
two sets of anvils independently. The strain inside are probed by X-ray diffraction, and
the strains are investigated by X-ray imaging techniques. This apparatus has the
advantage of handling relatively large samples, as opposed to the DAC technique.
However, the maximum pressure is limited to ~20 GPa (Nishiyama et al., 2008; Wang et
al., 2010; Kawazoe et al., 2010), corresponding to a depth of about 570 km, which is only
the transition zone pressure.
In fact, the lower mantle lies below 660 km and occupies about 65 vol% of the
Earth and delineates the thermal evolution and dynamics of the Earth. Its deformation
behaviors have been difficult to study and few mineral physics constraints have been
obtained. As a result, most of the previous studies used either theoretical estimates via
some empirical relations (e.g., Karato 1981) or were based on the experimental results of
analog materials (e.g., Poirier et al. 1983; Karato and Li 1992; Li et al. 1996). Those
approaches seemed not able to provide valid constraints on the stress and strength
properties of lower mantle phases because of the limitation of pressure and lack of
accuracy of stress measurements.
9
To extend the experimental pressure range, scientist performed deformation
experiments in DACs (Kinsland and Bassett, 1977; Sung, et al., 1977; Bisschop, et al.,
2005; Miyagi, et al., 2009). The maximum pressure at which rheological studies were
performed was greater than 200 GPa (Mao, et al., 1998).
The advantage of DAC
apparatus is that it can reach very high pressures, i.e. more than 400 GPa (Ruoff, et al.
1990), which allow one to perform deformation studies at pressures extended from upper
mantle to the core. The recent breakthroughs in using X-ray transparent gaskets coupled
with synchrotron radiation as the source of incident X-ray allows one to study the
strength and elasticity of silicates, oxides and metals that cover most of the important
phases within the Earth (Duffy et al., 1995; Kavner et al., 2003; 2002; Merkel et al.,
2002, 2003; Wenk et al., 2005; Shieh et al., 2004). This work takes advantages of the
well developed DAC technique and synchrotron radiation to investigate a MgO-NaCl
composite at deep mantle pressure.
1.5 NaCl, Mixture of NaCl and MgO
1.5.1
Sodium Chloride
NaCl is widely used as a test material due to its special properties such as simple cubic
structure, low strength, and clear diffraction pattern (Decker, 1971; Popescu et al., 2011).
Moreover, NaCl is also very popular in the high pressure studies. It serves as pressure
medium, pressure standard and insulation material for high pressure-temperature
experiments (Decker, 1971; Brown et al., 1999; You et al., 2009; Tateiwa et al., 2009;
Sakai et al., 2011).
At high pressures, NaCl transforms from rock salt structure (B1 phase, space group
Fm m) at about 30 GPa and room temperature to CsCl structure (B2 phase, space group
Pm m) (Bassett et al., 1968; Sato-Sorenson, 1983; Yagi et al., 1983; Heinz and Jeanloz,
1984; Sata et al., 2002; Nishiyama et al., 2003; Xiao et al., 2006; Ono, 2008). The
coordination numbers of cations change from 6 to 8 and the sample volume is reduced
around 5% (Bassett et al., 1968; Sato-Sorensen, 1983). The crystal structures of B1 and
B2 are shown in Figure 1.3.
10
Figure 1.3. Crystal structure of NaCl B1 and B2 phases. (a) B1 structure; (b) B2
phase. Green balls are Na atoms, blue balls are Cl atoms. B1 phase has a rock-salt
structure with 6 coordination numbers. B2 phase has a CsCl structure with 8
coordination numbers.
When NaCl serves as pressure transmitting medium, the strength of B1 phase
increases as pressure goes up (Meade and Jeanloz, 1988; Funamori, 1994; Weidner et al.,
1994). This may affect the hydrostatic environment. Consequently, the accuracy of
pressure determination may be also affected. Therefore, there is a strong need to evaluate
the stress state and stress development of NaCl under high pressure conditions.
Especially, the strength of B2 phase is still poorly understood.
1.5.2
Magnesium Oxide
MgO, with simple cubic structure (rock salt structure), is geologically related to an end
member of (Mg0.9, Fe0.1)O ferropericlase. Ferropericlase (Mg, Fe)O is believed to be the
second most abundant component (~20-30 vol%) of the Earth’s lower mantle (Ringwood,
1991; Helffrich and Wood, 2001). Its rheological properties can greatly affect the
convection and dynamics of the deep Earth due to two important reasons. First,
comparing properties of two main constituents of the Earth’s lower mantle- perovskite
and ferropericlase, the strength of ferropericlase is likely weaker, and therefore it may
accommodate most of the external strain and play a controlling role for the deformation
mechanism of the Earth’s lower mantle (Yamazaki and Karato, 2002). Second,
11
ferropericlase is likely to have a high elastic anisotropy (Tommaseo et al., 2006), which
will contribute to the understanding of seismic anisotropy in the Earth’s lower mantle
(Karki et al., 1997). The mechanical and elastic behavior of ferropericlase greatly
depends on the rheological and elastic properties of its end member MgO as it takes up
90 mol % in ferropericlase. Therefore strength of pure MgO end member at the
conditions relevant to the Earth’s mantle is of central importance for deep mantle
convection.
Moreover, MgO is widely used as a pressure standard in high pressure studies
(Zha et al., 2000). The equation of state of MgO is well constrained from a great number
of static compression measurements (Mao and Bell, 1979; Duffy et al., 1995; Fei et al,
1999; Devaele et al., 2000; Speziale et al., 2001). The elasticity of MgO has also been
studied extensively by Brillouin scattering (Sinogeikin and Bass, 1999; Sinogeikin et al.,
2000, Zha et al., 2000), and ultrasonic measurements (Isaak et al., 1989; Chen et al.,
1998). The shear strength of MgO has been investigated at high pressures and high
temperatures (Paterson and Weaver, 1970; Kinsland and Bassett, 1977; Meade and
Jeanloz, 1988; Duffy et al., 1995; Merkel et al., 2002; Uchida et al., 1996, 2004; Weidner
et al., 1994). However, studies of MgO mixed with other phases are very limited
(Kinsland and Bassett, 1977; Li et al., 2007). Therefore, it is very important to investigate
the deformation of MgO mixed in other phases, with a large contrast in strength, such as
NaCl.
1.5.3
Mixture of NaCl and MgO
NaCl has a bulk modulus of 23.8 GPa (Sata et al., 2002), while MgO has a bulk modulus
of 160.2 GPa (Speziale et al., 2001). The large contrast in elastic and mechanical
properties of NaCl and MgO makes them excellent analogue materials for modeling
(Mg,Fe)SiO3 perovskite (Pv) and (Mg, Fe)O ferropericlase (Fp), major assemblage in the
lower mantle. Such study is of great significance because perovskite and ferropericlase
are dominant phases in the lower mantle, which has great influence on mantle convection
and dynamics of the Earth. However, conducting experiments directly on the mixture of
perovskite and ferropericlase is extremely challenging. Perovskite phase is difficult to be
synthesized in powder form with a grain size in the range of 1-3 microns. Besides, the
12
iron content of the Pv and Fp phase are difficult to be consistent. The variations of the
iron content in the Pv and Fp phase hinder us to proceed the synthesis of the starting
material at this stage. On the other hand, MgO and NaCl have clear diffraction patterns,
large contrast of mechanical properties, and widely used in high pressure community.
Therefore, NaCl mixed with MgO was chosen as a good analog material for modeling
strength and deformation for mantle assemblages.
That is to say that NaCl and MgO mixture can be used as a good analog for the
complex Earth materials. With the large strength contrast, NaCl represents the relatively
soft material and MgO stands for a relatively strong material. Most parts of the Earth’s
interior are regarded as composed of aggregate materials. However, a large number of
experiments focus on single phase. Although it is a natural starting point, studying
mixtures are of essential as composite materials are more representative and may behave
differently from single end members. On the other hand, experiments on Earth minerals
may not be practical due to their rarity or the extreme experimental conditions required.
Therefore, NaCl –MgO mixture is proposed to serve as a good example for deformation
study of the rheology of the deep Earth.
1.6 Objectives and structure of this thesis
In this study, pure NaCl, MgO-NaCl mixture with 1:3, 1:2 and 1:1 volume ratio were
examined by synchrotron X-ray diffraction at beamline X17C, National Synchrotron
Light Source, Brookhaven National Laboratory. A radial geometry of DAC sample
position was chosen to yield deformation of differential stress (lower bound of yield
strength), micro-strain, phase transformation, and elasticity. The proposed goals of this
work are listed below:
(1) To explore the deformation mechanism of single phase and single phase in a
composite environment and a mixture material.
(2) To understand the strength of a mixture relates to its single-phase end members.
(3) To explore how phase transition exerts an influence on strength property of a
material.
13
(4) To evaluate how the phase boundary of its end member is affected by different
volume ratio of the mixed phases.
(5) To study the micro-strain within the sample and define the boundary between
elastic deformation and plastic deformation.
(6) To evaluate elastic properties of single phase, mixture, and single phase in a
mixed environment.
(7) To construct the strength model with different ratios of the MgO-NaCl mixture
materials.
In this work, bulk stress (lower bound of yield strength), stress on different planes of
NaCl, phase transition of NaCl B1 and B2 phases, elastic constants, microscopic
deviatoric strain of NaCl and MgO are reported.
The layout of this thesis is listed in Table 1.1. Chapter 1 provides a general
introduction of deformation study on materials. Chapter 2 describes the instrumentations
used in this thesis. Chapter 3 depicts the theory that back up this thesis. Chapters 4 to 7
constitute the body of the thesis, and are focused on pure NaCl, MgO mixed with NaCl at
1:3, 1:2 and 1:1 respectively. Lastly, Chapter 8 summarizes the general conclusions of the
thesis and outlines the future work.
14
Table 1-1 Summary staring materials, experimental methods, and pressure ranges
for the NaCl and MgO-NaCl mixture study in this thesis
Sample
Volume
Ratio
Maximum
Pressure
Method
Remarks
(GPa)
NaCl
pure
43
EDXD, Raman
MgO-NaCl
1:3
57.6
ADXD
MgO-NaCl
1:2
43.1
ADXD
MgO-NaCl
1:1
44.4
ADXD
stress, strain, elasticity,
phase transition
stress, elasticity,
phase transition
stress, strain, elasticity,
phase transition
stress, strain,
phase transition
15
Chapter 2
2
Experiments and Instruments
High pressure experiments were conducted at beam line X17C, National Synchrotron
Light Source, using a diamond anvil cell (DAC). Two types of X-ray diffraction were
employed in this study: energy-dispersive X-ray diffraction (EDXD) and angle dispersive
X-ray diffraction (ADXD). Both EDXD and ADXD studies used a radial geometry of the
DAC to study strength and elasticity of the mixture of NaCl and MgO samples (i.e. NaCl,
MgO-NaCl (1:1), MgO-NaCl (1:2), MgO-NaCl (1:3)). In this chapter, basic principles of
DAC, fundamentals of synchrotron radiation, powder X-ray diffraction and theory of
Raman spectroscopy and experimental methods were introduced.
2.1
The Diamond Anvil Cell
In high pressure science, pressure can be generated by two popular ways - dynamic and
static compression. The dynamic compression uses shock wave method and the static
compression uses piston-cylinder device, multi-anvil press and/or diamond anvil cell
(DAC). DAC gains its popularity due to its high transparency to a wide range of
electromagnetic radiation, visually observable the effects of pressure, and high strength to
produce ultra-high pressure above four megabars (400 GPa) ((Ruoff, et al. 1990), which
can attain the P-T range of Earth’s deep interior (Figure 2.1).
16
Figure 2.1. Pressures achieved by modern static compression methods and the
corresponding pressures at depth in the Earth (Geotherm), DAC-diamond anvil
cell; LVP-large volume press; RHDAC-resistive heating diamond anvil cell;
LHDAC-laser heating diamond anvil cell; together with the approximate P-T ranges
for planets.
There are several types of DACs, such as Mao-Bell type, Merrill-Bassett cell, and
customer-made symmetric DAC. In this study, symmetric DACs were adopted for
several different experiments and its geometry is shown in Figure 2.2. Two gem quality
diamonds are mounted to two tungsten carbide seats corresponding to piston and cylinder
of the DAC, respectively. Pressure is produced by mechanically tightening four screws to
force two diamonds opposing each other, further more the sample between two diamonds
is compressed. The force required even for the highest attainable pressure is not large,
due to a small culet size of the diamond. The culet size of diamond anvil determines the
highest pressure it could reach. Normally the smaller the size, the higher pressure it could
produce. The culet size used in this study generally was 300 μm, which allowed us to
achieve a pressure to about 60 GPa.
17
Figure 2.2. Schematic of diamond anvil cell. (a) compress-screws pushing diamonds
against each other to produce pressure; (b) indicates sample chamber
Diamonds are the most important parts in a DAC, due to their core feature of
generating high pressures. The rest of the cell serves to align the working tips, to keep
this alignment under loading and to sustain the load. Thus, knowing diamonds’ properties
and choosing proper diamonds for different experiments are necessary. Generally,
diamonds are classified into two main types: type I diamonds contain nitrogen atoms as
their main impurity, commonly at a concentration of 0.1%; type II diamonds have no
measurable nitrogen impurities. A further subdivision is introduced to make diamonds
into the following categories: type Ia has isolated nitrogen atoms; type Ib is of aggregated
nitrogen atoms; type IIa is clear diamond; type IIb shows the occurrence of boron
impurities. Type I diamonds have two strong absorption regions in infrared (IR) spectra,
which are around 2000 cm-1 and 1000-1350 cm-1. Type II diamonds have a clean
spectrum below 2000 cm-1 allowing IR measurements of the sample within the DAC.
Therefore, type I diamonds are suitable for Raman spectroscopy, while type II diamonds
are good for IR spectroscopy. At very high pressures it has been demonstrated that type I
diamonds with platelet nitrogen aggregates are more resistant to plastic deformation and,
therefore, may offer the best anvil material for diamond anvil cells operating in the Mbar
region (Mao et al. 1979).
The design of the anvils is a crucial factor for the maximum pressure a DAC can
achieve. Generally two types of cutting are widely used: the modified brilliant cut and
Drukker cut (Figure 2.3). Comparing to a brilliant cut diamond, the Drukker cut diamond
18
has an enlarged table area, an increased anvil angle, while without a highly stressed
shoulders. These modifications can strengthen the diamond anvil and sustain greater
applied loads. In this thesis, type Ia diamonds with Drukker cuts were employed for
synchrotron X-ray diffraction and Raman spectroscopy.
Figure 2.3. Types of diamond cutting. (a) Brilliant (b) Drukker types
Another important component of DACs is gasket. Without gasket, DAC fails at
around 10 GPa (Eremets, 2006). Gasket with a hole in the center serves as sample
chamber, building a lower pressure gradient and supporting the tips of the anvils.
Different types of gasket made by materials such as stainless steel, tungsten, rhenium,
beryllium, and boron-epoxy are used for different types of experiments. In this study,
beryllium and boron-insert beryllium gaskets were employed in synchrotron radial X-ray
diffraction experiments (Figure 2.4), because of their X-ray transparent properties while
stainless steel gaskets were applied for pressure gradient experiments conducted in-house
using Raman spectroscopy.
19
Figure 2.4 A beryllium gasket in between two diamonds tips
2.2 Synchrotron X-ray Radiation
When charged particles travel in a curved orbit at a relativistic speed, the tangentially
emitted electromagnetic radiation is called synchrotron radiation. It is extremely intense
over a broad range of wavelengths from the infrared to the visible and ultraviolet, and
into the soft and hard X-ray parts of the electromagnetic spectrum. Ever since its first
discovery at a General Electric (GE) laboratory in 1947 (Elder et al. 1947), synchrotron
radiation has become a premier research tool for the study of matter in all varied
manifestations, quickly developed from first generation to third generation.
A synchrotron facility (shown in Figure 2.5) is mainly composed by the following
parts:
An electron gun generates a large number of free electrons. Electrons are then
being accelerated by a Linac function to a speed close the speed of light, followed by a
process called injection which can be depicted by injecting electrons into a vacuum
chamber. Such a vacuum chamber (made by metal) called booster ring, the speedy
electrons then circulated inside the metal tube. The radio frequency cavity system (RF),
which acts on the circulating electrons and restores the energy that lose through the
emission of electromagnetic radiation. The bending magnets bend the trajectory of the
electrons and force them to circulate in a closed orbit. The insertion devices, such as
wigglers and undulators, are inserted into straight sections of the storage ring. They
further modify the electron trajectories form straight line, and thereby induce emission of
20
additional synchrotron radiations. The focusing magnets, which fine tunes the electron
beam trajectory to keep the electrons within a narrow range of a defined path. The beam,
which is the electromagnetic radiation, exits into the user’s experimental chambers.
Figure 2.5. Schematic view of a synchrotron radiation source. The injectionaccelerating system (Linac+Booster synchrotron) is inside the storage ring which is
actually like a polygon with the bending magnets at their vertices. Bending magnets,
BM, provoke the deflection of the electron trajectory and as a consequence produce
the synchrotron radiation which escapes forwardly. Insertion devices, ID, can be
positioned in the straight sectors to produce specific synchrotron light. The
radiofrequency cavity of the storage ring and those of the booster synchrotron are
also indicated.
A bending magnet produces a fan of radiation in the horizontal plane, like a
sweeping of search light. Several spatially separated experimental stations can be
installed from one bending magnet, each using a different part of the fan of radiation.
Wigglers and undulators are periodic devices, consisting of a series of magnets of
alternating polarities, which cause the electrons to curve back and forth. Therefore the
21
insertion devices can increase the intensity and brightness of the bending magnet
radiation by up to 10 orders of magnitude (Figure 2.6).
Figure 2.6. A historical graph shows the brilliance of X-ray sources from bending
magnet (BM), wiggler and undulator (Willmott, 2011).
The radiations emitted by successive wiggler magnets do not interfere with each
other, which are designed in practice. The total flux from a wiggler is the sum of the
emission from each magnet. Thus, wiggler generates incoherent radiation with a large
angle, allowing several branch lines to operate simultaneously by sharing a small portion
of the radiation. However, different from wiggler magnets, undulator has a smaller angle
of deflection of the electron trajectory from the straight line compared to mc2/E (m: mass;
c: light of speed; E: energy), and in this case the emissions from successive magnets
retain coherence and interfere with each other. The result of the interference is to enhance
the intensity in certain regions of wavelengths at the expense of other regions, and with a
sufficient number of magnets, the spectrum becomes concentrated to a single or a few
narrow, strong lines. For this reason, radiation from an undulator device can be used by
22
only one station at a time. The flux comparison of bending magnet, wiggler and undulator
is shown in Figure 2.7.
Figure 2.7. Comparative illustration of the generation process of synchrotron light
by a bending magnet (a) and by two different insertion devices (flat undulator b and
wiggler c): periodic, weak deflection of the electron beam in a flat undulator of
period length λp (λ: wavelength; p: period) and larger transverse oscillations of the
beam produced in a wiggler. Differences in photon beam collimation achieved for
the three magnetic elements are shown
Synchrotron radiation provides a very powerful tool for a broad range of sciences
extending in physics, biology, chemistry, Earth sciences, medicines etc. because of its
extraordinary properties such as: 1) the extremely high intensity allows a quick and
accurate measurement; 2) the high collimation ensures a great eventual resolution in
measurement due to its spatial precision. This advantage especially benefits high pressure
DAC study due to the small size of samples; 3) the continuous spectrum makes it
possible for conducting experiments with either white beam radiation or monochromatic
light. Because of these significant advantages, synchrotron radiation source has been
23
especially suitable for studying materials at small amounts and/or under extreme
conditions.
In this study, my experiments are mainly conducted at X17C, National
Synchrotron Light Source. The insert devices at X17C are wigglers, either with white
beam (EDXD) or with monochromatic beam (ADXD).
2.3 Powder X-ray Diffraction
By using a rich array of slits, collimators, focusing mirrors etc, the wide range of
synchrotron radiation can be dispersed into X-ray , ultra-violet, and infrared light into
different hutches for different types of studies. X-ray is high energy electromagnetic
radiation, with wavelengths between roughly 0.1Å and 100Å (i.e. in between γ-rays and
ultraviolet). Therefore, crystal structures can diffract X-rays due to a similar length scale.
X-ray powder diffraction is an important non-destructive method for determining a range
of physical and chemical properties of materials, such as phase identification, cell
parameters and crystal structure changes, crystallographic textures, crystalline sizes,
macro-stress and micro-strain. Different from a conventional lab X-ray source,
synchrotron X-rays are 1013 times more brilliant than that from an X-ray tube. The highenergy radiation is capable of penetrating the pressure container to probe high pressure
samples. The interaction between X-ray s and electrons of atoms in the samples, results in
a constructive spectrum when the path difference between two diffracted rays equals an
integral number of the wavelength of the incident X-ray. This selective condition is
described by the Bragg equation:
n=2d sin
where n is the order of the diffraction,  is the wavelength of (elastically) scattered
radiation, d is the spacing of the lattice planes in a sample, and  is the angle of specular
reflection with respect to these planes. The diffraction angle, defined as the angle
between the incident primary beam and the diffracted beam, is therefore equal to 2.
Diffraction experiments record scattered X-ray intensity as a function of either 2at a
fixed wavelength (ADXD) when monochromatic beam is used or as a function of energy
24
at a fixed 2when the white beam source is used EDXD. In this study, both EDXD and
ADXD were employed.
In ADXD (Figure. 2.8), the incident X-ray beam is monochromatized at a fixed
wavelength. Based on Bragg equation, nλ=2dsinθ, d-spacings are determined from the
observed θ angles. A single-crystal monochromator is used to select a small segment of
energy, typically several tenths of KeV, for high resolution ADXD. The highly
collimated monochromatic X-ray beam impinges upon a polycrystalline material in the
high pressure vessel and produces a number of diffraction rings at various 2θ angles
which are collected by an area detector such as an imaging plate or a charge-coupled
device (CCD). Two-dimensional data provide a wealth of information of the samples,
such as reliable peak intensity, crystallinity, stress, strain, texture and preferred
orientation. The high resolution of ADXD is crucial for many kinds of studies such as
splitting of diffraction lines for phase transition in crystals, multiple phase identifications,
high-order parameters of P-V-T equations of state, and second-order phase transitions.
Figure 2.8. Schematic angle dispersive X-ray diffraction (ADXD) setup using a DAC
at a radial geometry
EDXD (shown in Figure 2.9) uses the entire X-ray energy spectrum of
synchrotron radiation. The collimated polychromatic (white) X-ray beam passes through
the high pressure vessel and impinges on the specimen. The polychromatic diffracted
beam is collected by a solid-state detector fixed at a constant 2θ angle which disperses the
diffracted photons in the energy spectrum. Interplanar distances can be calculated using
25
observed peak energies, E: d=
. where h=6.626×10-34 J·s is the Planck’s constant.
The advantages of EDXD with a point detector compared with ADXD used an area
detector are a minimal requirement of accessible angle as well as high spatial resolution
along the primary X-ray beam path. The small opening of the cell for X-rays access
leaves available support of the anvils for reaching maximum pressures. Because both
primary and diffracted beams can be carefully aligned, the detector will only collect
signals from the sample centered area with or without limited contribution from the
gasket. In such a way, small samples and samples with low diffraction intensity can be
studied with minimal interference. This can also be crucial for amorphous materials at
high pressures, which yield very broad diffraction patterns comparable to the background
signal from non-Bragg scattering of the diamonds or the amorphous boron epoxy gasket.
By utilizing the entire X-ray spectrum of the polychromatic beam, EDXD is highly
efficient. The time resolution provided by the technique is an additional advantage for
high pressure-temperature experiments and for kinetic studies. The principal
disadvantage of EDXD is the lower energy resolution relative to ADXD.
Figure 2.9. Schematic energy dispersive X-ray diffraction (EDXD) setup using DAC
at a radial geometry
2.4 Raman Spectroscopy and Ruby Fluorescence
2.4.1
Raman Spectroscopy
When photons of a single wavelength (monochromatic light, in this case is a laser)
interact with matter, the in-elastically scattered phenomenon is called Raman Effect.
26
With Raman scattering, the incident photons interact with matter and their energy is
either shifted up or down. The photon interacted with the electron cloud of the bond
functional groups exciting an electron into a virtual state. The electron then relaxes into
an excited vibrational or rotational state. This causes the photon to lose some of its
energy and is detected as Stokes Raman scattering. This loss of energy is directly related
to the functional group, the structure of the molecule, the types of atoms in that molecule
and its environment. Therefore, Raman spectroscopy can provide a finger print analysis
on the vibration or rotation of materials. With a transparent diamond window, materials
under extreme pressure in a DAC could be studied by Raman spectroscopy.
In this study, a user-customized Raman system was combined with ruby
fluorescence method used to determine ruby pressures in a DAC. The schematic diagram
of the Raman system is shown in Figure 2.10. A beam of monochromatic light with a
wavelength of 514.32 nm is directed into the sample. Most of the light passes straight
through, but some is scattered by the sample. The scattering signals from the sample spot
are collected by a set of optics and sent through a spectrometer for analysis. Wavelengths
close to the laser line, due to elastic Rayleigh scattering, are filtered out while the rest of
the collected signals are dispersed onto a liquid-nitrogen cooled charge-coupled device
detector. The instrument control and data collection are performed by the software
WinSpec. The pressure can be obtained directly from the shifts of the ruby fluorescence
peak.
27
Figure 2.10. Layout of a Raman system used in this study for high pressure ruby
fluorescence measurements.
2.4.2
Ruby Fluorescence Scale
The ruby fluorescence method for pressure measurement, which involves measuring the
shift of the R1 luminescence band of ruby (A12O3 with Cr3+ impurity), was originally
proposed by Forman et al. in 1972 for its chemical inert and pressure dependence. Later,
this method was applied routinely in DAC study. It is a convenient, precise method for
determining the pressure.
Laser can excite ruby’s R1 and R2 lines and the doublet lines are intense and
sharp which is suitable for high pressure DAC experiments. With pressure increasing, the
lines shift toward the red end of the spectrum (Figure 2.11). Significantly, pressure could
be detected in situ using only a very small ruby crystal (comprising only 1 % of the
available volume) as the internal pressure sensor.
28
R1
3000
2500
18.79 GPa
2000
R1
1500
1000
692
10 GPa
694
696
698
700
702
704
706
708
710
Figure 2.11. Ruby R-line spectra at pressures of 10 and 18.79 GPa. R-lines shift to
the high frequency direction as pressure increase.
The shift of the ruby R1 line with pressure can be fit to the equation 2.1:
(2.1)
With A=1904 and B=7.665, where  is the peak wavelength of the ruby R1 at zero
pressure value and is line at high pressure value (Mao, et al. 1986).
2.5 Sample Preparation
Sample preparation in a DAC experiment plays an important role, because good sample
configuration is essential to a successful experiment.
In this study, four starting materials which are pure NaCl, mixture of MgO and
NaCl at 1:3, 1:2 and 1:1 volume ratio, were studied respectively.
In NaCl experiment, a symmetric diamond anvil cell with a pair of 300 μm culet
size diamonds was adopted. An X-ray transparent beryllium gasket pre-indented to 25 μm
thick with a hole of 100 μm in diameter was placed between two diamonds tips, served as
the sample chamber. All parts of the DAC were well aligned before loading samples.
NaCl powder (99.999% purity) from Alfa Aesar was mechanically ground to size of a
29
few microns using mortar and pestle. To eliminate moisture incorporated during grinding
in air, the NaCl sample was put into a furnace at ~120°C for at least 8 hours. Then the
sample was packed into the gasket hole and no pressure medium was loaded in order to
develop stress inside the sample. A tiny amount of gold powder was compressed between
two diamonds without gasket in a diamond anvil cell to form a gold foil. Lastly, gold foil
with square shape in ~20 μm size was picked and placed at the center of diamond tip,
serving as pressure standard and position marker (Figure 2.12).
In NaCl and MgO 1:1 mixture experiment, MgO powder (99.999% purity) from
Alfa Aesar was used as a starting material. Then MgO and NaCl powder were weighed in
1:1 volume ratio using Mettler Toledo balance on the basis of known densities. Next, the
mixed sample was mechanically ground for at least 2 hours. Acton was employed during
grinding for the purpose 1) to reduce the heat produced during grinding 2) to help two
sample powders well mixed. Due to the MgO and NaCl mixture exposure to the air, the
mixed sample was placed in a furnace at temperature of 120 °C for overnight drying out.
Then the mixture was loaded into the sample chamber, a 120 μm hole in a beryllium
gasket. Finally a gold foil was loaded on top of the sample and the cell was sealed.
The same sample preparation procedure was applied to MgO and NaCl mixture in
1:2 and 1:3 ratio.
Figure 2.12. Example of sample configuration for the radial X-ray diffraction study
in a DAC. NaCl sample is shown by white color and a small piece of yellow shiny
gold foil lies in the center of beryllium sample chamber.
30
Chapter 3
3
Theories
When applying stress to a material, firstly the stress may be built up and later
deformation may occur, causing volume or shape change. With regard to the deformation
of a small sample under extreme high pressure, such as deformations in a diamond anvil
cell, it can be represented by variations of d-spacings and/or grain sizes. Those changes,
in this study, were detected by X-ray diffraction method combined with a series of
theories. There are mainly three different methods of measuring deformation properties in
this work: lattice strain theory, pressure gradient method, and peak broadening profile.
Details are discussed as below in this chapter.
3.1 The Lattice Strain Theory
Lattice strain theory has been widely discussed elsewhere (Singh et al., 1993, 1994,
1998; Duffy et al., 1999, 2007), here it is presented a brief explanation. The sample is
subjected to a uniaxial stress given by two opposed diamonds. Since no pressuretransmitting medium was used, significant non-hydrostatic stress was produced in the
sample chamber. Upon compression, the pressure is distributed symmetrically but
decreasing from the loading axis to the edge of the sample. Meanwhile, the pressure
gradient is also developed in the sample. X-ray diffraction is sensitive to d-spacing
changes within the crystallites. Thus, the stress component of the sample could be
estimated by analyzing the X-ray diffraction data. The DAC with sample inside was put
on the sample stage, and the diffraction geometry is shown in Figure 3.1.
31
Figure 3.1. Schematic illustration of diffraction geometry in a radial X-ray
experiment in the diamond cell, σ3 is the axial stress while σ1 is the radial stress, ψ is
the diffraction angle.
The stress state at the center of the compressed specimen in the diamond anvil cell
can be expressed as (Singh, 1993),
(3.1)
In equation (3.1), σ3 and σ1 are the axial and radial stress respectively, while σp=
(σ1+σ1+σ3)/3 is the mean normal stress (hydrostatic pressure), Dij is the deviatoric stress
( i.e., the stress deviated from hydrostatic stress), and t is the differential stress, a lower
bound of yield strength on a Von Mises yield criterion, defined as:
(3.2)
where τ is the shear strength of the sample and Y is the yield strength.
The strains (eD) produced by the deviatioric stress Dij in terms of d-spacings can be
expressed as:
(3.3)
32
where dm(hkl) is the measured d-spacing under stress condition σ and dp(hkl) is the dspacing resulting from the hydrostatic component of stress.
Semi-empirical expression for the measured d-spacing, dm, can be written as a function of
ψ, an angle between the diamond loading axis and diffraction plane normal,
(3.4)
where Q (hkl) is defined as
(3.5)
where GR (hkl) is the aggregate shear modulus for the crystallites under the condition of
constant stress across grain boundary (Reuss condition, α=1). GV is the shear modulus
under iso-strain condition (Voigt limit, α=0).
The linear relationship between dm and 1-3cos2 ψ from equation (3.4) can be used to
derive Q(hkl) and hydrostatic dp under iso-stress condition (α=1), therefore, the
differential stress can be expressed as
(3.6)
For the cubic systems, GR (hkl) and Gv (hkl) have a relation with the single crystal elastic
compliances Sij as follows (Sing, 1993)
(3.7)
(3.8)
where the elastic compliance S=S11-S12-1/2S44 and
(3.9)
33
Equations (3.5, 3.7- 9) suggest a linear relationship between Q(hkl) and 3Γ, in which
under Reuss condition its intercept (m0) and slope (m1) can be expressed as
(3.10)
(3.11)
In cubic system, the isothermal bulk modulus KT can be expressed as
(3.12)
From equation (3.10-12), all three independent elastic compliances for the cubic crystal
can be deduced as
(3.13)
(3.14)
(3.15)
Relations between single crystal elastic compliances Sij and elastic constants Cij are given
by (Nye, 1985)
(3.16)
(3.17)
(3.18)
34
Based on the relationship between Cij and Sij (3.16-18), the single crystal elastic constants
can be expressed by:
(3.19)
(3.20)
(3.21)
The bulk modulus KT and its variation with pressure can be calculated using finite-strain
equation in the Eulerian coordinate system (Poirier, 2000):
(3.22)
where,
(3.23)
(3.24)
(3.25)
where
0
is the density of sample at ambient pressure,
is the density of sample under
pressure. K0T and K’0T are the isothermal bulk modulus and its pressure derivative at
ambient conditions, respectively. Values of isothermal bulk modulus and its pressure
derivative for MgO and NaCl were obtained from the fit of a third-order BirchMurnaghan equation of state to the experimental volume-pressure data at ψ=54.7° under
hydrostatic condition.
35
3.2 Pressure Gradient Method
The pressure gradient method was well explained by Meade and Jeanloz (1988 a, b).
Here a brief interpretation is provided.
The pressures are influenced by the radial distance from the centre to the edge of
the sample. Highest pressure distributed around the centre while lower pressure existed at
the edges. That is to say, the pressure profiles are almost symmetrical along the axis of
the centre (Figure.3.2). The stress in the diamond anvil cell is approximately axially
symmetric, the force balance for the stresses can be written as follow (Fung 1965):
σ
σ
σ
σ
σ
σ
(3.26)
σ
(3.27)
where z is parallel to the loading axis, r is the radial distance from the center of the
sample, and
is perpendicular to r in the plane of the culet face. Because of the axial
symmetry condition, none of the stress components vary in the θ direction (
), and the only nonzero stresses are
,
, and
. With this coordinate system,
all stresses are symmetric about the axis r = 0 and the mid-plane z = 0. To prevents the
third terms,
and
must vanish at r = 0 by symmetry. With two
approximations, one can integrate (3.26) and obtain an expression for
σ
σ
where
.
(3.28)
based on the symmetry
. The shear stress the sample
exerts on the diamond is then be expressed by,
σ
σ
(3.29)
where h/2 is half of the sample thickness in μm. The integration shows that at a particular
radius the magnitude of
increases away from 0 at mid-plane (z=0) to its maximum
36
value at the culet and sample interface. In the pressure range of our experiment we expect
that since the shear stresses (
. the radial gradients of the normal stresses
are all approximately equal to
, thus the shear stress calculation can
be written as (3.30)
σ
(3.30)
Figure 3.2. Schematic plot of stress distribution inside a diamond anvil cell
(modified from Meade and Jeanloz, 1988). The ruby spheres (red dots) are
distributed across the sample chamber and are used for pressure determination.
However, this pressure gradient method may be controversial. Reports suggested
that this method will exaggerate the σ value (Jing et al., 2007) but some supported the
idea as the results measured by X-ray diffraction and pressure gradient method are
roughly similar (Kavner and Duffy, 2001, 2007). The thicknesses of the sample cannot be
measured in-situ. Therefore, systematical errors could introduce incorrect estimation of
the yield strength. Because of this, further assessment for this method will be needed.
37
3.3 Peak Width Theory
There are some factors that may affect the observed peak profile, such as instrumental
effect, crystallite size, micro-strain, solid solution in-homogeneity. Solid solution inhomogeneity can be avoided as a result of well grinding. With regard to the instrumental
effects, it will not be discussed here since it is contributed from machine. Therefore here I
mainly grain size, L, and strain, ε into consideration.
Scherrer (1918) found that the peak broadening is in reverse correlation with the
grain size, and can be described in the equation below:
λ
(3.31)
θ
where Bs is the peak broadening caused by grain size, λ is the X-ray wavelength, θ is the
Bragg angle, K is Scherrer constant that usually is taken to be 0.9.
The second factor here I discuss is in-homogeneous micro strain. Based on previous work
(Stokes and Wilson, 1944; Williamson and Hall, 1953), the line width Bd due to
distortion/strain (ε) is described by:
ε
θ
(3.32)
where constant C depends on the assumptions made concerning the nature of the
inhomogeneous strain, but is typically ≈ 4 or 5.
In summary, total line broadening can be caused by either elastic strains or small grain
sizes. Hence, we can express the total peak broadening as:
(3.33)
λ
θ
ε
θ
(3.34)
where B represents full width at half maximum (FWHM) measured from the sample,
is due to the grain size contribution and
is due to crystal distortions contribution.
38
This expression holds for Lorentzian peak profile, however, many peaks have a Gaussian
profile, therefore equation (3.33) can be written into,
(3.35)
Rewriting expression 3.35 into 3.36 as below
(3.36)
By plottingB2 (2θ) cos2 θ versus 4 sin2θ, grain size can be derived from intercept, and
micro-strain can be obtained from the slope.
Expressions 3.31-3.36 are well fitted with mono-chromatic ADXD measurement.
However, regarding to the white beam EDXD method, the equations should be expressed
as below:
The peak broadening is in the correlation with grain sizes can be rewritten into (Gerward
et al., 1976; He et al., 2006):
(3.37)
θ
where h is Pplanck constant, c is the light speed; 2θ is the angle between diffraction line
and Ge detector, which is at a fixed value.
Referring to the peak broadening contributed by distortion/micro-strain, it can be written
into expression 3.38:
ε
where E stands for diffraction energy.
The total peak broadening caused by elastic strains grain sizes can be expressed as:
(3.38)
39
(3.39)
By plotting B(E)2 as a function of 4 E2, grain size can be derived from intercept, and
micro-strain can be obtained from the slope.
40
Chapter 4
4
Strength and Elasticity Study on NaCl
NaCl is widely used as a test material because of its special properties such as simple
cubic structure, low strength, and clear diffraction pattern (Decker, 1971; Popescu et al.,
2011). Moreover, NaCl gain great popularity in high pressure studies. It serves as a
pressure medium, pressure standard and insulation material for a broad spectrum of highpressure and high-temperature research (You et al., 2009; Tateiwa et al., 2009; Decker et
al., 1971; Brown et al., 1999; Sakai et al., 2011). However, as pressure increases, shear
stress may develop inside the pressure medium itself and therefore affect the hydrostatic
environment (Meade and Jeanloze, 1988; Funamori, 1994; Weidner. et al., 1994).
Consequently, the accuracy of pressure determination may also be affected. Therefore,
there is a strong need to evaluate the stress state and stress development inside NaCl
under high pressure conditions, especially for the poorly known high pressure B2 phase,
to provide better insight into the suitability of NaCl for future applications, and more
experiments need to be carried out to well constrain the phase boundary.
NaCl is the simplest ionic crystals. It is well known that NaCl with a rock salt
structure (the B1 phase, space group Fm m) transforms into a CsCl structure (the B2
phase, space group Pm m) at pressure range of 27 to 32 GPa, based on XRD
experimental data at room temperature (Bassett et al., 1968; Sato-Sorenson, 1983; Yagi
et al., 1983 Heinz and Jeanloz, 1984; Nishiyama et al., 2003; Sata et al., 2002). However,
immense theoretical studies predicted the B1-B2 transformation pressure for NaCl from
25 GPa (Ono, 2008) to 30.7 GPa (Perez-Albuerne and Drickamer, 1965). Therefore, both
experimental data and theoretical calculations show discrepancies for the NaCl B1-B2
transition pressure.
Shear strength of NaCl B1 phase has been studied to the upper pressure limit of its
structural stability zone (Auten et al., 1973; Kinsland and Bassett, 1977; Meade and
Jeanloz, 1988; Weidner et al., 1994; Funamori et al., 1994). However, very little is
known about the pressure effect on the mechanical strength of NaCl B2 phase. There are
41
no data available on the strength of NaCl at pressures higher than 42 GPa (Meade and
Jeanloz, 1988). Moreover, the only reported values for the strength of NaCl B2 phase are
obtained from indirect measurements using pressure gradient technique which may result
in incorrect estimation of the strength (Jing et al., 2007; Kavner and Duffy, 2001).
Therefore, the problem of the shear strength of NaCl at high pressures is very important
because it is a popular pressure medium and may affect the accuracy of high pressure
experiments. Therefore, there is necessary call for the investigation of the differential
stresses (i.e. the lower bound of yield strength under a Von Mises yield criterion) of NaCl
at high pressures.
In this study, the strength of NaCl was studied under non-hydrostatic conditions in a
diamond anvil cell at pressures across the NaCl B1-B2 transition at room temperature
using radial EDXD technique. The proposed goals of this study are: (1) to explore
strength of NaCl B1 and B2 phase; (2) to find out the validation region of NaCl as good
pressure transmitting medium candidate; (3) to study the deformation mechanism of
different planes under high pressure; (4) to define the boundary between plastic and
elastic deformation by strain - stress curve; (5) to investigate the elasticity of NaCl to
mantle pressure.
4.1 Experiment
NaCl powder (99.999% purity, Alfa Aesar) was mechanically ground by mortar and
pestle to sizes of 1~3 μm. Next, NaCl sample was placed in a drying oven at temperature
of 120°C for 8 hours to eliminate moisture during grinding in the air. The sample was
loaded into the sample chamber, a 100 μm size hole drilled through a beryllium gasket
which was pre-indented to 26 μm. A ~15 μm thin gold foil was placed in the center of
diamond culet as a pressure standard and position marker. No pressure medium was
loaded in order to develop non-hydrostatic stress inside the sample.
Radial energy-dispersive X-ray diffraction experiment was performed at beamline
X17C, National Synchrotron Light Source, Brookhaven National Laboratory. Solid state
germanium detector (Figure. 4.1) was used to collect high pressure diffraction data in this
study and was calibrated by a gold standard fixed at 2θ=11.994°. The incident X-ray
42
beam was collimated by a pair of Kirkpatrick-Baez mirrors to a size of 25 by 25 μm2,
perpendicular to the loading axis of the diamond anvil cell. X-ray beam was directed
through both the beryllium gasket and the sample. X-ray diffraction patterns were
collected over the angle range of 0 to 90°. Each diffraction pattern was collected about 10
minutes. To minimize the uncertainty of the pressure determination during each
compression step, the sample was allowed to relax for at least 30-120 minutes for stress
relaxation.
Figure 4.1. EDXD experiment set up in a radial geometry at beamline X17C, NSLS,
Brookhaven National Laboratory.
4.2 Results and Discussions
4.2.1 X-ray Diffraction Patterns
High pressure EDXD experiment on NaCl was gradually compressed up to 43.7 GPa
with a pressure step of 3-5 GPa. The diffraction patterns were collected at ψ angle of 0°,
24°, 35°, 45°, 55°, 66°, and 90°, respectively. Therefore, totally seven to eight diffraction
43
patterns were acquired at one single pressure. The representative diffraction patterns for
the NaCl sample performed at a pressure of 29.8 GPa are shown in Figure 4.2. It can be
clearly observed that the diffraction peaks shift to lower energy as the ψ angles increase,
as evidence shown by two arrows in Figure 4.2 for Au 311 between ψ=90° and 0°. The
smallest energy value is found at around ψ=90°. Inferred from the equation
and
λ=2dsinθ (where E is energy, h is the Planck constant, c represents the speed of light and
λ the wavelength of X-ray), energy value has an inverse relation with d-spacing.
Therefore, small energy values represent the large d-spacings. At ψ=90°, the samples
have largest d-spacings means a minimum stress condition. On the other hand, it also
suggests that the sample would be under the largest stress at 0°. Therefore, the diffraction
peaks shift to lower energy as the ψ angle is increased, indicating the strain decrease as
the diffraction plane normal approaches the minimum stress axis.
29.8 GPa
Au 311
Au 220
B2 110
B2 100
8000
B1 200
10000
0°
6000
90°
Intensity
4000
66°
55°
2000
45°
0
35°
20
25
Be
30
35
40
Energy (KeV)
45
50
55
B1 422
B1 420
Be
Au 222
Au 311
B1 400
Be
B1 222
B2 200
B1 311
24°
B1 220
Be
-4000
Be
Au 200
B1 111
Au 111
-2000
60
0°
65
Figure 4.2. Representative X-ray diffraction patterns of NaCl B1 phase obtained at
different ψ angles at 29.8 GPa. B1: NaCl B1 phase; B2: NaCl B2 phase; Au: gold;
Be: beryllium gasket. The ψ angles were labeled with each pattern. The arrows
indicate the shifts of the peak under different stress condition.
44
4.2.2 Phase Transformation
The diffraction pattern at ψ=54.7º is regarded as at hydrostatic condition on the basis of
equation 3.4 (chapter 3) since 1-3cos2θ=0. Hence, diffraction patterns at ψ=55º are
regarded as close to hydrostatic condition. Figure 4.3 shows those diffraction patterns
collected at ψ=55º under different pressures. It can be clearly observed that all sample
phases (except beryllium peaks) shift to higher energy direction with increasing pressure.
The sample was subject to compression and resulted in reduction in d-spacings.
Eventually phase transformation occurred to response to further compression. Figure 4.3
shows that NaCl begins a phase transformation from B1 to B2 at 29.8 GPa and is
completed at 32.3 GPa. This result agrees well with previous studies, falling into the
pressure range of 27 -32.3 GPa (Bassett et al., 1968; Sato-Sorensen, 1983; Heinz and
Jeanloz, 1984; Nishiyama eta l., 2003; Sata et al., 2002). New peaks such as B2 100 and
Au 222
Au 311
B2 210
B2 200
Au 220
B2 111
Be
Be
B2 100
8

Au 200
3
10x10
B2 110
110 starts to show up at 29.8 GPa, accompanied by fade out of B1 phase.
43.7
38.9
Intensity
6
32.3
B2
4
29.8
26.9
2
-2
20
30
Energy (KeV)
40
50
Be
15.2
B1 420
B1 331
Au 222
Be
Au 311
Be
B1 222
Au 220
B1 222
Be
Be
B1 220
Be
Au 111
B1 200
B1 111
Au 200
Be
20.1
0
10.2
60
Figure 4.3. Diffraction patterns of NaCl collected at hydrostatic condition up to 43.7
GPa. The arrows show the emergence of NaCl B2 phase started at 29.8 GPa and
accomplished fully at 32.3 GPa. B1 denotes NaCl B1 phase, B2 denotes NaCl B2
45
phase, Au denotes gold and Be denotes beryllium. Pressure is labeled with each
pattern at the right hand side of the Figure.
4.2.3 Strength Study
The variations of d-spacing with 1-3cos2ψ are shown in Figure 4.4. For both NaCl B1 and
B2 phases, a linear relationship of observed d-spacings with 1-3cos2ψ is observed at
different pressures for all cases. Q value is derived from the slope based on equation (3.4)
and can be used towards future calculations since it is the ratio of differential stress to
shear modulus. The hydrostatic d-spacing, dp(hkl) value can be obtained at ψ=54.7° or 13cos2ψ=0. The corresponding hydrostatic pressure was determined from gold peaks based
on the reported equation of state of Fei et al. (2007). In Figure 4.4, the slopes of B1 111
are steeper than B1 200, suggesting a larger value of differential stress over shear
modulus.
3.00
10.2
B1 111
2.65
B1 200
2.60
10.2
2.55
15.2
15.2
d_spacing (Angstrom)
d_spacing (Angstrom)
2.95
20.1
2.90
2.85
20.1
2.50
26.9
26.9 2.45
29.8
29.8
-2.0
-1.5
-1.0
-0.5
2
1-3cos 
0.0
0.5
1.0
2.40
-2.0
-1.5
-1.0
-0.5
2
1-3cos 
0.0
0.5
1.0
46
29.8
1.51
B2 200
B2 110
32.3
2.13
32.3 1.50
d_spacing (Angstrom)
d spacing (Angstrom)
2.12
2.11
1.49
38.9
38.9
2.10
1.48
43.7
2.09
1.47
43.7
2.08
-2.0
-1.5
-1.0
-0.5
2
1-3cos 
0.0
0.5
1.0
1.46
-2.0
-1.5
-1.0
-0.5
2
1-3cos 
0.0
0.5
1.0
Figure 4.4. d-spacings as a function of 1-3cos2ψ for NaCl, B1 111 200 and and B2
110 and 200. The pressure is labeled next to each fitted lines.
The ratio of differential stress to shear modulus, t/G, is directly obtained from the
slope of the d-spacing as function of (1-3cos2ψ) with assumed Reuss conditions (isostress) in the sample. In Figure 4.5, my results on the values of t/G fall into the range of
0.009(6)-0.013(6) from 10.2 to 43.7 GPa. For NaCl B1 phase, the t/G of NaCl shows
very gentle curve, which suggests the increase of differential stresses almost equivalent to
that of shear modulus. Across B1-B2 phase transition, the t/G value has an abruptly
decrease at 29.8 GPa, and immediately climbs up from 0.00006 to 0.0034. This indicates
that strength weakening happened along the phase transition from B1 phase to B2 phase.
At high pressure (i.e. B2 phase regime) the increase of the differential stress is greater
than that of shear modulus, as inferred from the steep ascending slope. It is noted that the
scale of differential stress to shear modulus of B2 phase rises faster than the
corresponding values for NaCl B1 phase. The open circles with crosses reported by
Akhmetov (2008), both B1 and B2 phase show higher values than this study. Such
difference may arise from different sample configuration: Akhmetov (2008) used NaCl
and MgO mixture, and I used pure NaCl.
Diffenrential stress/shear modulus
47
10
-2
10
-3
NaCl B1 phase
this study
Akhmetov, 2008
10
-4
10
-5
NaCl B2 phase
this study
Akhmetov, 2008
12
16
20
24
28
32
36
Pressure (GPa)
40
44
48
52
Figure 4.5. The ratio of differential stress to shear modulus as a function of pressure
for B1 and B2 phase of NaCl.
Figure.4.5 shows the ratio of differential stress to shear modulus (Q factors) for
different planes of NaCl. The Q factors of NaCl B1 111, 222 and 422 are much higher
than those of 200 and 400, which means d-spacing of B1 111 changed faster than that of
B1 200 from ψ=0º to ψ=90º. On one hand, it means NaCl B1 111 is more sensitive to
stress variation and thus probably will lead to a larger pressure variation. On the other
hand, d-spacing of 200 shows less sensitive to pressure change, and may serve as a better
pressure indicator. Gold pressure scale based on d200 of gold changes more than that of
d111, which is also more consistent with other gold peaks and thus the pressures
calculated by Au 111 are more reliable. Our results show a good agreement with those of
Funamori et al. (1994). Therefore, the pressure derived from NaCl 200 is more reliable
than that of NaCl 111 due to geometry of Γ and a negative S value (Funamori et al.
1994).
48
3.0x10
-3
222
422
Differential stress/Shear Modulus
2.5
111
2.0
1.5
420
1.0
200
0.5
400
0.0
15
20
Pressure (GPa)
25
Figure 4.6. Ratios of differential stress to shear modulus value of B1 phase as a
function of pressure for different planes of NaCl.
Alternatively a quick decrement in d-spacing of NaCl{111} planes is observed
(Figure 4.4), which indicates NaCl{111} planes have the figuration that stacking on each
other and forming texture if plastic deformation initialed; that is the bonds of atoms on
the planes {111} are hard to break. In this way one can imagine that actually all planes
have the inclination to glide along {111} planes, as if the slip got activated. Slip is the
process by which plastic deformation is produced by a dislocation motion. Not all
experiments performed by the DAC reach plastic deformation region. Therefore, in this
experiment, if NaCl undergoes elastic deformation within B1 phase region, the slip
systems will not be activated yet. However, if {111} planes are arranged in the same
direction and pattern, texture will be developed. Since the sample bears the largest
uniaxial stress along loading axis (stresses along axial direction are much larger than
radial direction), eventually {111} will be distributed perpendicular to the axial direction
to sustain the increasing stress. Wenk et al. (2009) studied the texture pattern of NaCl
under different strain rate and reported that {111} are favored planes when the strain rate
is greater than
. This study of stress on different planes may also provide
clues of texture information. Regarding to deformation mechanics, NaCl{111} planes are
49
of higher yield point than NaCl{200} planes. Therefore, {200} planes will deform first
under high pressure condition.
With the input of shear modulus, the differential stress can be obtained according to
equation (3.6) while assuming that the sample was under iso-stress conditions. The NaCl
shear modulus value was obtained from Brillouin scattering data (Bass, et al. 2006;
Lakshtanov, 2007). Figure 4.7 shows differential stress on different planes supported by
NaCl. Solid symbols were obtained in this experiment, open symbols were reported by
Funamori et al. (1994). For NaCl B1 phase, largest differential stresses were supported
by (111) and (222), whereas (200) and (400) sustain the lowest stress. As differential
stress represents the lower bound of yield strength under Von Mises condition, therefore,
NaCl B1 111 yields last and 200 planes are responsible for the onset deformation under
high pressure.
Figure 4.7. The differential stress supported by different planes of NaCl as a
function of pressure. Solid symbols and lines are from this study whereas open
symbols and dashed lines are from Funamori et al. (1994)
50
Figure 4.8 shows the differential stresses of B1 phase increasing from 0.22(6) to
0.38(6) GPa, indicating that the differential stress of B1 phase undergoes small changes
under compression. Nonetheless, along the B1-B2 phase transformation the differential
stress decreases significantly to 0.002(6) GPa around applied pressure of 29.8 GPa
(Figure 4.7).When the B2 phase transition is complete, the differential stress supported
by NaCl B2 phase increases remarkably and eventually reaches 0.8(6) GPa at the
maximum applied pressure of 43.7 GPa. It is noted that B2 phase has higher differential
stress than that of B1 phase, and a significant strength weakening occurs across the B1B2 phase transformation zone. Generally, our data are slightly lower than those of
previous reports, i.e. Kinsland and Bassett (1977), Meade and Jeanloz (1988), Funamori
et al. (1994), and Weidner et al. (1994) as shown in Figure 4.8. However, Kinsland and
Bassett (1977) and Weidner et al. (1994) obtained their NaCl data from a NaCl and MgO
mixture. The strong material of MgO may have significant influence on the soft material
NaCl in a mixed environment. Nevertheless, Funamori et al. (1994) reported embedded
NaCl in a mixture of amorphous boron and epoxy resin in their measurements. It is well
known that boron is a very strong material and thus their results may be affected by the
boron. Meade and Jeanloz (1988) measured the shear stress of NaCl by pressure gradient
method, which may be problematic. Recent reports suggested that this method will
underestimate the strength value (Jing et al., 2007). However, some supported the idea
that the strength results measured by X-ray diffraction and pressure gradient way are
roughly similar (Kavner and Duffy, 2001, Duffy, 2007). To evaluate the pressure gradient
method, we measured the shear stress of NaCl by pressure gradient method (solid triangle
shown in Figure 4.8). It turned out that the pressure gradient method provides similar
values of those of lattice strain theory method. However, the abrupt drop of strength at
~43.8 GPa for the pressure gradient method has been observed for SiO2 in the report of
Meade and Jeanloz (1988 a, b). The interpretation of phase transition is not supported by
this NaCl study as there is no phase transition found for B2 phase at this pressure regime.
One plausible explanation is the thickness of gasket at pressure above 42 GPa introduced
large uncertainty, as the gasket will sustain large part of stress. Therefore, the shear stress
determined by pressure gradient method may be biased at high pressure. The effect of
gasket should be taken into consideration.
51
Differential Stress (GPa)
1.5
Kinsland and Bassett, 1977
Meade and Jeanloz, 1988
Weidner et al., 1994
Funamori et al., 1994
NaCl (P gradient, this study)
NaCl (XRD, this study)
1.0
0.5
0.0
10
20
30
Pressure (GPa)
40
50
Figure 4.8. Comparison of differential stress supported by NaCl at different
pressures obtained from this study with previous published reports (Kinsland and
Bassett, 1977; Meade and Jeanloz, 1988; Weidner et al., 1994; Funamori et al., 1994).
Solid green symbols and green line are NaCl data from RXRD, whereas solid pink
symbols are NaCl data from pressure gradient method.
4.2.4 Strain Study
As pressure increases, sample undergoes a large amount of stresses, causing strain to
develop inside the sample. The change of micro-strain and grain size can affect the full
width at half maximum (FWHM) of peaks. By peak broadening method described in
chapter 3 on the basis of expression 3.39, [B (E)]2 as a function of E2 were plotted, where
B stands for FWHM and E denotes energy. The micro-strain derived from the slop shown
as in Figure 4.9.
52
Figure 4.9a shows stress as a function of strain for NaCl B1 phase, with five
corresponding pressure steps at 10.2, 15.2, 20.1, 26.9 and 29.8 GPa, respectively. From
the last two data points, stress doesn’t increase much while strain keeps raising,
suggesting a yield point and a transition from elastic to plastic deformation. Figure 4.9b
shows the microscopic deviatoric strain distribution of NaCl at different pressures,
obtained by equation 3.39 (chapter 3). Strain of B1 phase increases linearly as pressure
increase, indicating an elastic deformation. Alternatively, strain of B2 phase can be fit to
a nearly constant value from 32.3 to 43.7 GPa, which indicates a plastic deformation
stage.
25x10
-3
25x10
-3
NaCl B1
20
Strain
Strain
20
15
10
5
5
0.28
0.32
Stress (GPa)
(a)
0.36
NaCl B2
15
10
0.24
NaCl B1
10
15
20
25
30
Pressure (GPa)
35
40
(b)
Figure 4.9. a) Strain as a function of stress for NaCl B1 phase; b) strain as a
function of pressure for NaCl B1 and B2 phases. The green symbols and lines
represent NaCl B1 phase and purple symbols stand for B2 phase.
4.2.5 Elasticity Study
The single crystal elastic moduli were obtained from equation 3.19-21 (detail in chapter
3), together with the theoretical results and Brillouin scattering data (Figure.4.10).
Generally, our NaCl B1 values obtained from lattice strain theory (experimental data) are
broadly consistent with previous published reports and our own theoretical calculations to
27 GPa. C11 values are in agreement with theoretical calculation and Brillouin scattering
53
data (Bass et al. 2006). C12 values are slightly lower than previous data. C44 data are
higher than theoretical and experimental data. A decreased value of C11 was observed at
29. 9 GPa, due to a phase transition to B2 structure and also a transition from elastic to
plastic deformation (Figure.4.10). In addition, the elastic constants of B2 phase are not
consistent with theory, a piece of evidence that B2 phase is within plastic deform regime
(Figure.4.10).
400
Kiefer (unpublished)
Xiao et al.,2006
Whitefield et al.,1976
Bass et al.,2006
NaCl B1, this study
NaCl B2, this study
Elastic Constants
300
C11 (NaCl B2)
C11 (NaCl B1)
200
C12 (NaCl B2)
C12 (NaCl B1)
100
C44 (NaCl B2)
C44 (NaCl B1)
0
0
4
8
12
16
20
24
28
32
Pressure (GPa)
36
40
44
48
52
Figure 4.10. Comparison of single crystal elastic constants of NaCl obtained from
the lattice strain theory and theoretical calculations. Solid circles represent B1 phase,
solid diamonds indicate B2 phase. Open symbols are from previous studies (Xiao et
al., 2006; Bass et al., 2006; Whitefield et al., 1976).
4.3 Conclusion
Radial X-ray diffraction of NaCl at pressure up to 43.7 GPa show phase transformation
from B1 to B2 starting at 29.8 GPa, and complete at around 32.3 GPa. The strength of
54
NaCl shows very gentle increase with pressure, which indicates that NaCl is a very good
pressure-transmitting medium at pressure below 30 GPa (Figure.4.7). However, the
differential stresses that can be supported by NaCl abruptly increased for B2 phase
(Figure. 4.7) and is no longer a “soft” pressure medium at higher pressures.
Elastic and plastic regions were defined using the peak broadening method. The
deformation of NaCl B1 remains in elastic region, whereas B2 phase undergoes a plastic
deformation instead. The elastic constants of B1 phase calculated by lattice strain theory
show agreement with previous published reports to 27 GPa, as beyond 27 GPa NaCl is
subject to plastic deformation.
Comparison between our differential stress and previous published reports show that
the strong material may affect soft materials in a mixture environment. Thus the
differential values reported by Funamori et al. (1994), Kinsland (1977) and Weidner et
al. (1994) are therefore higher than this study. Furthermore, our pressure gradient
measurements on NaCl show it is in broad agreement with lattice strain theory results.
The shear stress values are not reliable above 43.8 GPa due to the uncertainty of gasket
thickness.
Differential stresses supported by different planes indicate that {111} planes of
NaCl B1 phase are the strongest planes and are associated with the glide system and
texture development. NaCl B1 {200} are the weakest planes and may deform first under
high pressure. NaCl 200 can better represent the pressure scale when NaCl served as
pressure standard. d111 varies much faster than d200. Therefore, the {111} planes are easy
to stack on each other, but hard to break, thereafter forming a texture along {111}
direction. In summary, the differential stress study on different planes may be useful as a
guide to define the texture information, which is of significant meaning for future study.
55
Chapter 5
5
Strength and Elasticity Study on MgO-NaCl (1:3)
Mixture
In this experiment, MgO-NaCl (1:3) mixture was studied up to 57.6 GPa in a symmetric
DAC by angle dispersive X-ray diffraction, at X17C, NSLS. The goals of the
experiments are: (1) to compare stress and elasticity with pure NaCl (see chapter 4)
obtained by EDXD measurements (i.e. a different environment from this ADXD
measurements); (2) to find out the differences and similarities between the two
experimental setups; (3) to evaluate ratios of differential stress to shear modulus in two
different samples; (4) to investigate how the differential stresses of a soft material (NaCl)
is affected when in contact with a strong material (MgO); (5) to evaluate deformation
mechanisms on different crystal planes of the aforementioned mineral phases; (6) to
obtain elastic constants of NaCl B2 phase which are poorly studied.
5.1 Experiment
All experiments were conducted at X17C, NSLS, Brookhaven National Laboratory.
Similar experimental procedures as Chapter 4 were used. But, in some cases
modifications and changes of experimental setup were applied to improve the quality of
the acquired data. Therefore, only a brief description of the experimental procedure is
provided.
NaCl and MgO were weighed in a 3:1 volume ratio using Mettler Toledo balance
according to their densities. Next, the mixture was well ground for at least 2 hours in an
acetone environment, to reduce the heat produced during grinding and to ensure the
homogeneity of the mixture. Due to the hydrophobic nature of MgO and NaCl mixture,
moisture may be retained with the mixed sample when exposed to air. As a result, the
mixed sample was placed in a furnace to dry out at a temperature of 120 °C for overnight.
A symmetric DAC with a pair of 300 μm culet size diamonds was used as high pressure
apparatus. An X-ray transparent beryllium gasket was pre-indented to 29 μm thick and
then a 120 μm hole was drilled in the center of the gasket by electrical discharging
machining (EDM). A gold foil with a size of 24 by 20 μm2 serving as pressure standard,
56
was loaded on top of the sample and then the cell was sealed. The cell was mounted on a
custom-made sample holder designed particularly for symmetric DAC (Figure 5.1). The
X-ray beam was collimated to a size of 25 by 25 μm2, perpendicular to the loading axis of
the DAC in radial geometry. A monochromatic X-ray wavelength was fixed at 0.4066 Å
for this ADXD measurement. X-ray diffraction 2-D images were collected by a charged
couple device (CCD) detector. The diffraction rings can be recorded in an angle of more
than 90 degrees at ψ angle. Therefore, much information can be obtained using this type
of setup. Each pressure step was collected for a minimum of 10 minutes. To minimize the
uncertainty of the pressure determination after each compression step, the sample was
allowed to relax for at least ~30minutes before data collection.
Figure 5.1. ADXD experiment set up in a radial geometry at X17C, NSLS.
Due to the space limitation inside the hutch, the CCD detector was placed off the
center position. Therefore, the 2-D image was not shown in full rings but half circles
were recorded instead. Figure 5.2a shows the 2-D image of CeO2 Calibrant. Using the
Fit2D program, the 2-D rings can be transformed into caked patterns with information of
two theta at X axis and azimuthal angle ψ at Y axis (Figure 5.2b).
57
(a)
(b)
Figure 5.2. (a) A typical 2-D image of CeO2 recorded in this work; (b) a typical
caked diffraction image deduced from 2-D rings.
5.2 Results and Discussions
5.2.1 2-D Images
High pressure ADXD experiment on MgO-NaCl (1:3) composite was compressed up to
57.6 GPa. Each pressure increment was ~3 GPa, and 2-D images at different pressures
were collected (Figure 5.3 a, c, e). Every 2-D image was caked by the Fit2D program
into several lines with different two theta value representing different d-spacings of NaCl,
MgO or gold (Figure 5.3 b, d, and f). The Y axis denotes arbitrary azimuthal angles
which from 0° to 180°. In Figure 5.3a and b, 2-D image was collected at 5.26 GPa, NaCl
B1 111 and 200 peaks are labeled aside. The lines (peaks at different azimuthal angle) are
straight without obvious wavy shape which is interpreted as no stress in the sample. As
pressure increases to 31.7 GPa, a transition pressure between B1 and B2, and the
diffraction lines exhibit obvious wavy features, suggesting the buildup of significant
stress (Figure 5.3 d). At 57.6 GPa (Figure5.3 f), the wavy shapes are even enhanced,
indicating a higher stress environment within the samples.
58
Figure 5.3. 2-D image collected at 5.26, 31.7 and 57.6 GPa respectively. The
corresponding caked images are shown at certain two theta angles to emphasize the
development of the stress upon the sample.
59
5.2.2 1-D Diffraction Patterns
After data collection, each 2-D image was sliced into several sections at a five- degrees
step and each five-degree section was integrated into 1-D diffraction patterns (Figure.
5.4a, b, and c). Stacking the diffraction patterns together explicitly show peaks shift with
azimuthal angle. The peak shifts with azimuthal angle allow us to quantitatively analyze
the stress condition within the sample. In Figure 5.3, 2-D images at 5.26 GPa, 31.7 GPa
and 57.6 GPa were presented for low pressure, phase transition and high pressure
condition. Here the corresponding 1-D diffraction patterns are shown in Figure 5.4. The
x-axis denotes two theta angle and the diffraction patterns are labeled with corresponding
Miller indices. In this study, 1-D patterns sliced at every five degree angles were stacked
from 0° to 180° on the top, which are different from seven patterns collected at 0°, 24°,
35°, 45°, 55°, 66° and 90° in chapter 4 (Figure 4.1). As pressure increased, the peaks
move strongly with azimuthal angle and show increasing stress conditions (e.g. Au 200 at
57.6 GPa). In the case of Au 200 peaks, the smallest two theta value was found at 90°,
suggesting the biggest d-spacing. Because θ has a inverse relation with λ based on
Bragg’s law
. A large d-spacing indicates a low stress environment. Therefore,
it can be inferred that the sample would be under the smallest stress at 90° and largest
stress at 0° and 180°.
60
2500
5.26 GPa
B1 222
Be
B1 311
Be
B1 220
Au200
Au111
B1 200
Intensity
B1 111
1500
MgO200
Be
2000
1000
180
500
90
0
-500
0
11.0
12.0
Two Theta
B1 200
B1 111
B2 100
BeO
MgO 111
31.7GPa
13.0
14.0
15.0
Au 200
10.0
BeO
9.0
B2 110
1500
8.0
Au_111
7.0
180
Intensity
1000
500
90
0
-500
0
7.6
8.0
8.4
8.8
9.2
9.6
10.0
10.4
Two Theta
10.8
11.2
11.6
12.0
12.4
61
Au 220
1000
MgO 220
B2 200
Be
Be
Au 111
BeO
B2 100
1500
Au 200
57.6 GPa
Intensity
180
500
90
Be
B2_110
0
-500
0
8.0
9.0
10.0
11.0
12.0
13.0
Two Theta
14.0
15.0
16.0
17.0
Figure 5.4. 1-D diffraction patterns collected at 5.26, 31.7 and 57.6 GPa respectively.
B1: NaCl B1, B2: NaCl B2, Au: gold, Be: beryllium.
5.2.3 Phase Transformation
Phase transformation from B1 (rock salt structure) to B2 (CsCl structure) starts at 29.8
GPa and is complete at 32.3 GPa for pure NaCl sample (see chapter 4). In this experiment,
NaCl B2 110 peak appeared at 29.4 GPa and B2 100 peak shown up at 31.7 GPa. Phase
transition completed at 36.7 GPa (Figure 5.5). As pressure increases, peaks shift to large
two theta angles (small d-spacings). MgO 111 and 200 peaks are overlapped with
beryllium oxide (BeO) and beryllium (Be) peaks and therefore cannot be used for further
analysis. In such a case, Au 111, 200 and 220 peaks were used to determine pressure in
this work.
62
4000
57.6
3000
Au 220
B2 200
Au 200
MgO 200
B2 110
Au 111
BeO
B2 100
ψ=55°
Intensity
55.9
54.3
52.5
50.7
2000
48.3
45.7
B1 200
43.4
39.7
B2 110
36.7
31.7
29.4
27.9
24.3
21.5
8.0
9.0
10.0
11.0
12.0
Two Theta
13.0
14.0
11.7
8.08
5.26
B1 222
B1 311
B1 220
Au 200
MgO 200
BeO
BeO
Au 111
MgO 111
B1 111
0
B1 200
1000
15.0
16.0
17.0
Figure 5.5. Diffraction patterns of MgO-NaCl (1:3) mixture collected at hydrostatic
condition up to 57.6 GPa. Arrows show B1 to B2 phase transition started at 29.4
GPa and accomplished at 36.7 GPa.
Our phase transition boundary is found to be consistent with previous studies
(Bassett et al., 1968; Sato-Sorensen, 1983; Heinz and Jeanloz, 1984; Nishiyama et al.,
2003) and falls into the pressure range of 27 to 32.08 GPa (table 5.1). The variations may
be due to different experimental conditions and kinetic issues that extend the NaCl phase
transition to a relatively high pressure. In this experiment, to avoid the potential kinetic
problem that may affect the results at least one hour elapse on our samples was applied to
allow stress relaxation and phase transition development.
63
Table 5-1. Comparisons of NaCl B1 and B2 phase transition in different studies.
Published
Pressure (GPa)
at Start of Transition
Techniques
Bassett et al.
1968
30
XRD (DAC)
Yagi et al.
1983
27
XRD (DAC)
Sato-Sorensen
1983
29
XRD (DAC)
1984
29
XRD (DAC)
Author
Heinz and
Jeanloz
Visual
Li and Jeanloz
1987
30
observations
(DAC)
infrared
Hofmeister
1997
32
spectroscopy
(DAC)
Sata et al.
2002
32.08
XRD (DAC)
Nishiyama et al.
2003
29.3
NaCl (chapter 4)
This study
29.8
XRD (DAC)
MgO-NaCl (1:3)
This study
29.4
XRD (DAC)
XRD (Multi
anvil apparatus)
5.2.4 Strength Study
Reliable two theta values of the mixed MgO-NaCl sample can be obtained by PeakFit
program. Values of d-spacings can be calculated on the basis of Bragg’s law
.
The d-spacings generally decrease with pressures increase. Moreover, the d-spacings also
varied at same pressure step (Figure 5.6 a, c) because of the stress conditions are varied at
different locations of the sample. In general, the maximum d-spacing is found at 90° and
the minimum d-spacing values found at 0° and 180° (Figure 5.6 b, d), indicating that the
sample undergoes minimum stress at 90°. It can be seen by a sinusoid function shown in
d-spacing versus ψ angle plot. If all d-spacing values fit well with a smooth and
symmetric sinusoid curve, the consequent d-spacing versus (1-3cos2ψ) plot will show a
perfect linear relationship (Figure 5.6 b, d). The non-linearity in the dm(hkl) versus (1-
64
3cos2ψ) plots is attributed to the shift of the sample off from the center of the anvil face,
lack of parallelism of the anvil faces, and the presence of texture in the sample (Singh,
2000). In Figure 5.6, typical d-spacings as a function of (1-3cos2 ψ) plots of NaCl B1 and
B2 are presented. NaCl B1 111 peaks are collected from 5.26 to 31.7 GPa and NaCl B2
110 peaks are probed from 31.7 to 57.6 GPa. All of them show good linear fittings. Q
(hkl) value and hydrostatic d-spacing, dp, can be derived from the slopes and intercept of
the linear fit. Note that when the slope is steep, the bigger Q value will be obtained.
(a)
3.10
(b)3.10
5.26
5.26
3.05
3.05
8.08
d_spacing (angstrom)
d_spacing(angstrom)
8.08
3.00
11.7
2.95
B1 111
3.00
11.7
2.95
B1_111
2.90
2.90
21.5
21.5
24.3
2.85
24.3
27.9
2.85
27.9
31.7
2.80
0
20
40
60
80
100
120
Azimuthal Angle ψ
140
160
180
31.7
2.80
-2.0
-1.5
-1.0
-0.5
2
1-3cos Ψ
0.0
0.5
1.0
31.7
31.7
2.12
2.12
B2 110
(d)
(c)
36.7
39.7
2.08
43.4
2.06
45.7
36.7
39.7
2.10
d_spacing (angstrom)
d_spacing (angstrom)
2.10
B2 110
43.4
2.08
45.7
48.3
50.7
2.06
52.5
54.3
48.3
55.9
50.7
2.04
52.5
57.6
2.04
54.3
55.9
57.6
2.02
0
20
40
60
80
100
Azimuthal angle
120
140
160
180
2.02
-2.0
-1.5
-1.0
-0.5
2
1-3cos Ψ
0.0
0.5
1.0
65
Figure 5.6. Plot of d-spacing versus ψ angle shows sinusoid curves for NaCl B1 111
and B2 110. The corresponding d-spacing as a function of 1-3cos2 ψ shows linear fits
for both phases. Pressure values are labeled on the right vertical axis.
The ratio of differential stress to shear modulus, t/G, is directly obtained from the
slope of the d-spacing as function of (1-3cos2ψ) with assumed Reuss conditions in the
sample. This is directly from experimental data with no contribution from any theoretical
input. Figure 5.7 shows that the ratios of differential stress over shear modulus of 111 are
much larger than those of 200, which means d-spacing of 111 changed faster with ψ
angle than that of NaCl 200. On the other hand, variations of d-spacings of 200 indicate
less sensitivity to stress condition and may serve as a better pressure indicator. The values
of t/G fall into the range of 0.0008(6)-0.0055(6) at 5.26 - 57.6 GPa (Figure. 5.8). For
NaCl B1 phase, the t/G fitting shows gentle curve, which suggests that the increase rate
of differential stress is almost equivalent to the rise of shear modulus. Strength
weakening happened along the phase transition due to the volume change, from NaCl B1
phase to B2 phase. At higher pressure (i.e. NaCl B2 phase) the increase rate of the
differential stress is greater than the rise of shear modulus, which can be inferred from the
ascending steep slope. It can also be noticed that the scale of differential stress to shear
modulus of NaCl B2 phase rises faster than the corresponding values for NaCl B1 phase.
66
Differential Stress/Shear Modulus
5x10
-3
4
B2 100
B1 111
3
B2 110
B1 220
2
B1 200
1
10
20
30
40
Pressure (GPa)
50
Figure 5.7. Plot of differential stress over shear modulus as a function of pressure
for MgO-NaCl (1:3) mixture. Miller indices are labeled with each line. It shows that
the increment of differential stress of B1 111 over shear modulus is the greater than
other planes of B1 phase.
With the input of shear modulus, the differential stress can be obtained according to
equation 3.6 (chapter 3) assuming the samples under iso-stress conditions. The NaCl
shear moduli were obtained from Brillouin scattering data (Bass, 2006; Lakshtanov,
2007). In Figure 5.8, the differential stresses of NaCl at different planes exhibit same
tendency as Figure 5.7. For B1 phase, t111 is much larger than t200, which indicate that
{200} planes may deform first under high pressure, and then tend to glide on {111}
planes. For NaCl B2 phase, t100 is larger than t110, which means t110 will initially start the
deformation under high stress.
67
2.5
Differential Stress (GPa)
2.0
B2 100
1.5
1.0
B1 111
B2 110
B1 220
0.5
B1 200
8
12
16
20
24
28 32 36 40
Pressure (GPa)
44
48
52
56
Figure 5.8. Differential stress supported by NaCl at different planes at pressure up
to 57.6 GPa. NaCl B1 phase shows a lower differential stress than those of B2 phase,
indicating a better pressure medium than B2.
Figure 5.9 shows the comparison of the strength of NaCl in MgO-NaCl (1:3)
mixture, pure NaCl and previous published reports (Kinsland and Bassett, 1977; Meade
and Jeanloz, 1988; Funamori, 1994; Weidner, 1994). NaCl B1 phase shows the
differential stress from 0.43(6) to 0.61(6) GPa in the pressure range of 5.26 - 27.9 GPa,
indicating that the differential stresses of B1 phase underwent small. Nonetheless, at
27.9-31.7 GPa, around the B1-B2 phase transformation, the differential stress decreases
to 0.36 (6) GPa. When the NaCl B2 phase transition completed, the differential stresses
supported by NaCl B2 phase increased remarkably and eventually reached 2.36(6) GPa at
the maximum applied pressure of 57.6 GPa. It is noted that B2 phase has higher
differential stresses than those of B1 phase, and a significant strength weakening occurs
across the B1-B2 phase transformation zone due to volume change. In chapter 4, the
relation between pressure and differential stress of pure NaCl can be written in (1) B1
phase: t = 0.009156P + 0.118154; (2) B2 Phase: t= 0.055198P - 1.636562. In this
experiment, the differential stress and pressure can be expressed by (1) B1 phase:
t=0.014751P+0.210062; (2) B2 phase: t=0.078432P-2.2685. Obviously, the t values of
68
NaCl in composite environment (mixed with MgO at 1:3 volume ratio) are much larger
than that of pure NaCl. On a average, the stresses of NaCl B1 in 1:3 mixture environment
are 65-75% larger than pure NaCl, and the stresses of B2 phase in 1:3 mixture
environment are 45-70% higher than that that of pure one. It can be concluded that the
strong material (MgO) might affect the soft material (NaCl) by lowering the porosity and
defect inside the sample.
2.5
Kinsland and Bassett, 1977
Meade and Jeanloz, 1988
Funamori et al., 1994
Weidner et al., 1994
NaCl (P gradient, this study)
NaCl (XRD, this study)
NaCl (XRD, 1:3, this study)
Differential Stress (GPa)
2.0
1.5
1.0
0.5
0.0
4
8
12
16
20
24 28 32 36
Pressure (GPa)
40
44
48
52
56
Figure 5.9. Comparison of differential stress supported by NaCl B1 and B2 phases
at pressure to 56 GPa. Solid symbols are from this study and open symbols are from
previous studies (Kinsland and Bassett, 1977; Meade and Jeanloz, 1988; Funamori
et al., 1994; Weidner et al., 1994)
5.2.5 Elasticity Study
The single crystal elastic moduli were obtained from equations 3.19-21 (see chapter 3).
The experimental data of NaCl in composite environment are compared with previous
experimental
measurements,
theoretical
results
and
Brillouin
scattering
data
69
(Figure.5.10). The red solid symbols represent elastic moduli of NaCl in the mixture
condition (MgO-NaCl in 1:3. Our NaCl B1 values obtained from lattice strained theory
(experimental data) are roughly consistent with previous published reports, though C11
values are slightly smaller than previous data. C12 and C44 data are higher than theoretical
and experimental data. In addition, the elastic constants of B2 phase are not consistent
with theory, maybe B2 was already underwent a plastic deformation.
400
Kiefer (communication)
Xiao et al.,2006
Whitefield et al.,1976
Bass et al.,2006
NaCl (1:3), this study
Elastic Constants (GPa)
300
200
C11 (NaCl B2)
C11 (NaCl B1)
C12 (NaCl B2)
100
C44 (NaCl B2)
C12 (NaCl B1)
C44 (NaCl B1)
0
0
4
8
12
16
20
24
28
32
Pressure (GPa)
36
40
44
48
52
56
Figure 5.10. Comparisons of single crystal elastic constants of NaCl obtained from
the lattice strain theory (NaCl in mixture), theoretical calculations and Brillouin
data (pure NaCl). Solid symbols are from this study. Open symbols and dashed and
dash-dotted lines are from previous published reports (Xiao et al., 2006; Bass et al.,
2006; Whitefield et al., 1976).
70
5.3 Conclusions
Radial ADXD measurement of NaCl mixed with 25% of MgO was performed at pressure
up to 57.6 GPa. ADXD measurements provide rich information within the same period of
collecting time compared to EDXD method. However, EDXD can provide better special
resolution than that of radial ADXD. In this radial ADXD measurement, high two theta
angles could not be observed due to a small ω angle. Phase transformation from B1 to B2
started at 29.4 GPa and completed at around 36.7GPa. Comparing to the completion of
B2 phase at 32.3 GPa for pure NaCl, MgO (strong material) may extend B1 stability. The
differential stresses of NaCl in composite environment are larger than that of pure NaCl.
They are 65-75% larger for B1 phase and 45-70% larger for B2 phase. Strong material
helps to strengthen soft material. For NaCl B1 phase, 200 planes will deform first at an
elevated pressure. For NaCl B2 phase, 100 and 200 planes are strong planes. Elastic
constants of NaCl B2 phase are not quite consistent with theoretical data. It may be due
to a plastic deformation developed in the B2 phase.
71
Chapter 6
6
Strength and Elasticity Study on MgO-NaCl (1:2)
Mixture
MgO-NaCl mixture with a ratio of 33.3 vol% MgO and 66.7 vol% NaCl were studied in a
DAC using radial ADXD method. The major purpose is to explore how the strength of
soft material being affected by strong material with a higher MgO volume ratio. The
second focus is to construct the strength model with MgO-NaCl mixture. Strain of NaCl
and MgO were investigated to determine elastic and plastic deform regime.
6.1 Experiment
The experiments were carried out at X17C, NSLS, Brookhaven National Laboratory.
MgO-NaCl (1:2) mixture was investigated in a radial geometry by angle dispersive X-ray
(ADXD) diffraction method. A diamond anvil cell serves as pressure apparatus. The
pressures were performed up to 43.1 GPa.
MgO powder (99.999% purity) and NaCl powder (99.999% purity) from Alfa
Aesar were weighed by Mettler Toledo balance into 1:2 volume ratio. The mixture was
mechanically ground to a few microns and was examined under microscope. Acetone
was employed during grinding to reduce the heat and to ensure the homogeneity of
mixture of two powder samples. Due to the MgO and NaCl mixture exposures to air, the
moisture may be incorporated. The mixed sample was placed in a furnace to dry out at a
temperature of 120 °C overnight.
A beryllium gasket with a 100 μm hole was put between the tips of two 300 μm
diamonds, serving as a sample chamber. A thin gold piece, acting as pressure scale, was
placed at the center of the anvil. The X-ray wavelength was fixed at 0.40653 Å. A
charge-coupled device (CCD) area detector was employed to acquire diffraction data.
72
6.2 Results and Discussions
6.2.1 1-D Diffraction Patterns
MgO-NaCl (1:2) mixture was gradually compressed to 43 GPa with a 2-3 GPa step. 2-D
X-ray diffraction images were obtained for a collection time of 10 minutes. Then the
Fit2D program was employed to deduce the 2D images into 1-D patterns at a five-degree
interval. These 1-D diffraction patterns with different ψ angles were stacked together, as
shown in Figure 6.1. The diffraction patterns are presented from ψ=0° to 180° at the top.
The Miller indices are also labeled with the peaks. The peaks labeled with “Be” are
contributed from beryllium gasket. It can be clearly seen that the different ψ angles, two
theta values of the sample varied but not for beryllium (Be) peaks. Moreover, it can be
obviously observed that NaCl B2 110 and Au 200 show strong curve-like shapes
compared to the unchanged Be peaks in Figure 6.1b. The smallest two theta value
occurred at around ψ=90°. The dashed lines in Fig 6.1b indicate the curve tendency.
Based on the Bragg law 2dsinθ=λ, two theta value has a inverse relation with d-spacing.
Therefore, small two theta values equal to large d-spacings. In other words, at ψ=90°, the
samples have largest d-spacings and is under the minimum stress condition. It can also
be therefore inferred that the sample would be under the largest stress at both 0° and 180°.
73
1500
Be
(a)
Be
Au 200
Be
B2 110
B2 100
1000
Au 111
43.1 GPa
Intensity
500
MgO 220
B2 200
180º
0
90º
-500
0º
8
10
12
Two Theta
14
16
(b)
Figure 6.1. (a) Diffraction patterns of MgO-NaCl (1:2) were collected at 43.1 GPa.
Miller indices are labeled with peaks. B2 denotes NaCl B2 phase, Be indicates
beryllium, Au represents gold. The ψ angles from 0° to 180° are labeled with peaks.
(b) The zoom-in plot with two theta at 10.5-12.5 shows that smallest two theta value
appears at ψ=90° and biggest values at both 0° and 180°. Dashed line shows two
theta changing tendency.
74
6.2.2 Phase Transformation
As pressure increases, sample peaks shift to higher two theta (i.e. small d-spacing)
direction, indicating that sample is subject to compression. The volumes of sample
become smaller in response to the new environment. Eventually at certain high pressure,
phase transition occurred and accompanied with structure change. As shown in Figure 6.2,
NaCl B2 110 peak first appeared at 30.7 GPa, accompanied by the fade of NaCl B1 peaks.
B1 200, the most intense peak for B1 phase disappeared completely at 38.8 GPa,
indicating the completion of phase transition. The total transition time from B2
emergence to the disappearance of B1 was 145 minutes.
MgO 220
B2 200
Be
Be
MgO 200/Be
B2 110
2000
Au 111
B2 100

Au 200
2500
43.1
B1
40.6
1500
Intensity
38.8
34.5
33.5
1000
30.7
29.7
Au 220
25.5
21.2
10.7
Au 220
MgO 220
Be
B1 222
Be
B1 220
Au 200
MgO 200
Au 111
MgO 111
B1 200
0
B1 111
Be
MgO 220
B1 222
26.8
500
3.8
7.0
8.0
9.0
10.0
11.0
12.0
Two Theta
13.0
14.0
15.0
16.0
17.0
Figure 6.2. Diffraction patterns of MgO-NaCl (1:2) mixture collected at hydrostatic
condition up to 43.1 GPa. The arrows show the emergence of B2 phase started at
30.7 GPa and accomplished fully at 38.8 GPa. Pressures are labeled at the right
vertical axis.
In addition to the peak shiftings, peak broadenings occurred simutaneously while
pressure increases. Peak broadening features can be easily observed in B1 200 and B2
110. At low pressure,i.e. 3.8 GPa, B1 200 shows sharp and narrow feature. As pressure
75
increasing, B1 200 peaks exhibit blunt and broad characteristics to reflect the stress
condition within the sample. Similarly, the high-pressure form of NaCl B2 110 peaks also
become broad at higher pressures. It is known that the grain size and micro-strain inside
the sample can affect peak shapes. Therefore, peak broadening can be used to investigate
information of grain size and micro-strain.
My results showed that phase transition boundary between NaCl B1 and B2 phase
are slightly varied in different MgO-NaCl mixture. In this study, the transition started at
30.7 GPa and completed at 38.8 GPa while for the pure NaCl it started at 29.8 GPa and
completed at 31.7 GPa. In the case of MgO-NaCl (1:3) mixture, NaCl B1 and B2 phase
transition started at 29.4 GPa and finished at 36.7 GPa. Table 6.1 showed the phase
boundary between B1 and B2 varied in different mixture.
Table 6-1 Phase boundary between B1 and B2 phase with different vol% of MgO
sample
B2 phase
B1 phase
appear (GPa)
disappear (GPa)
Volume ratio
NaCl
Pure
29.8 (6)
32.3 (7)
MgO-NaCl
1:3
29.4 (4)
36.7(3)
MgO-NaCl
1:2
30.7 (5)
38.8(5)
Peaks contributed by the gasket could overlap with sample peaks and hence affect
data analysis. To resolve this issue, background subtraction method was applied. First, a
sample image and a gasket image at same pressure were collected by same exposure time.
Second, gasket contributions were subtracted by Fit2D program to allow clean sample
peaks. Unfortunately, MgO 220 was confined by the limited two theta angles and is not
qualitatively good enough for differential stress calculation. Therefore, differential stress
of MgO will not be discussed in this chapter.
76
6.2.3 Strength of NaCl
Reliable two theta values and FWHM of the mixed MgO-NaCl sample can be obtained
by PeakFit program. Therefore, the differential stress, micro-strain and elastic constants
can be quantitatively investigated. In Figure 6.3, d-spacings of NaCl B2 110 as a function
of ψ angle are presented. The pressures, labeled at the vertical axis, were shown from
33.5 GPa to 43.1 GPa. The d-spacings generally decrease with pressure increase.
Moreover, the d-spacings also varied at same pressure step, clearly seen by a sinusoid
function. In general, the maximum d-spacing is found at ψ=90° and the minimum dspacing values found at 0° and 180°, indicating that the sample undergoes minimum
stress at 90°.
2.12
33.5
2.11
d_spacing (angstrom)
34.5
2.10
B2 110
2.09
38.8
2.08
40.6
2.07
43.1
0
25
50
75
100
ψ angle
125
150
175
Figure 6.3. Plot of NaCl B2 110 d-spacing versus ψ angle shows sinusoid curves
from 33.5 to 43.1 GPa. Pressure (hydrostatic pressure) is labeled with the curves.
The d-spacings are decreasing with pressure. The largest d-spacings are at around
90° for all pressures.
According to equation 3.4 (chapter 3), the observed d-spacings fit a linear function
with (1-3cos2ψ). The intercepts yield hydrostatic d-spacings, from which hydrostatic
pressure can be derived. The slopes reveal information of Q (hkl), the ratio of differential
77
stress over shear modulus. With input of shear modulus, the differential stresses can be
obtained. In Figure 6.4, a typical d-spacing as a function of (1-3cos2ψ) plot of NaCl B2
110 was presented. Linear functions were clearly displayed. NaCl B2 110 peaks are
collected from 33.5 to 43.1 GPa. If all d-spacing values well fit with a smooth and
symmetric sinusoid curve in Figure 6.3, the consequent d-spacing versus (1-3cos2 ψ) plot
will show a perfect linear relationship. The non-linearity in the dm(hkl) versus (1-3cos2ψ)
plot is attributed to the shift of the sample off from the center of the anvil face, lack of
parallelism of the anvil faces, and the presence of texture in the sample (Singh, 2000).
33.5
2.12
34.5
d_spacing (angstrom)
2.11
38.8
2.10
B2 110
40.6
2.09
43.1
2.08
2.07
-2.0
-1.5
-1.0
-0.5
2
1-3cos ψ
0.0
0.5
1.0
Figure 6.4. NaCl B2 110 d-spacings as a function of 1-3cos2ψ at pressure of 33.5-43.1
GPa. Linear fits can be clearly observed. Pressure is labeled with each linear fit. The
d-spacings decrease as pressures increase. Dashed line indicates the intercept and
also the hydrostatic condition.
The ratio of differential stress to shear modulus, t/G, is directly obtained from the
slope of the d-spacing as function of (1-3cos2ψ) with assumed Reuss conditions in the
sample. This is directly from experimental data with no contribution from any theoretical
input. Figure 6.5 shows that the shear strains of B1 111 are much larger than that of B1
200, which means d-spacing of B1 111 changed faster with ψ angle than that of B1 200.
78
A strength weakening occurred along the phase boundary. For B2 phase, differential
stresses over shear modulus of different planes show similar increment. However, B2 200
shows higher value than that B2 110, indicating d-spacing of B2 200 changed faster with
ψ angle than that of B2 110.
MgO:NaCl (1:2)
Differential Stress/Shear Modulus
4.5x10
-3
4.0
B1 111
B2 200
3.5
B1 220
3.0
2.5
B2 110
B1 200
2.0
1.5
4
8
12
16
20
24
28
Pressure (GPa)
32
36
40
Figure 6.5. Plot of differential stresses over shear modulus as function of pressures
for NaCl in MgO-NaCl (1:2) mixture. Miller indices are labeled with each line. It
shows that the increment of differential stress of B1 111 over shear modulus is
greater than other planes of B1 phase. For B2 phase, the B2 200 shows the highest
value than other planes.
With the input of shear modulus, the differential stress can be obtained according to
equation 3.6 (chapter 3) assuming the samples were under iso-stress conditions. The
NaCl shear modulus value was obtained from Brillouin scattering data (Bass, 2006;
Lakshtanov, 2007). In Figure 6.6, differential stress (lower bound of yield strength) of B1
111 is much larger than that of B1 200, indicating that {200} plane may deform easily
and {111} planes will yield lastly. For NaCl B2 phase, t100 is slightly larger than t110,
which means t110 will initially start to deform under high stress. Over all, at high pressure
79
or under the lower mantle condition, initial deformation of NaCl will be triggered along
B1 200 planes.
1.8
1.6
Differential Stress (GPa)
1.4
B2 200
1.2
1.0
0.8
B1 111
B2 110
0.6
B1 220
0.4
B1 200
0.2
4
8
12
16
20
24
28
Pressure (GPa)
32
36
40
Figure 6.6. Differential stress of NaCl B1 and B2 along different planes at high
pressures. Miller indices are labeled with each plane. NaCl B1 200 has the lowest
differential stress (yield strength), which means B1 200 may yield first under high
pressure.
In Figure 6.7, the differential stresses of MgO-NaCl (1:2) mixture are compared to
the strength of pure NaCl, MgO-NaCl (1:3) mixture and previous published reports
(Meade and Jeanloz, 1988; Funamori, 1994; Kinsland and Bassett, 1977; Weidner, 1994).
NaCl B1 phase of MgO-NaCl (1:2) mixture shows the differential stress from 0.26(6) to
0.91(6) GPa in the pressure range of 3.8 – 30.7 GPa. A strength softening occurred
around the B1-B2 phase transformation. The differential stress of NaCl B2 phase of
MgO-NaCl (1:2) mixture increases to 1.02(6) GPa at the maximum applied pressure of
43.1 GPa. In this experiment, the differential stress and pressure can be expressed by (1)
for NaCl B1 phase: t = 0.023923P + 0.166592; (2) for NaCl B2 phase: t = 0.099594P 2.810276. Obviously, the t values of NaCl in composite environment (MgO-NaCl 1:2)
are much larger than that of pure NaCl and NaCl mixed with 25% MgO (see chapter 4
and 5). In general, the stresses of NaCl B1 in 1:2 mixture environments are about 50-120%
larger than pure NaCl and 5-35% larger than NaCl in MgO-NaCl (1:3) mixture condition.
80
The stresses of NaCl B2 phase in MgO-NaCl (1:2) mixture environment are 90-160%
higher than that of pure NaCl, and 30-50% larger than that of NaCl B2 in MgO-NaCl (1:3)
mixture. It is therefore concluded that the strong material (MgO) can strengthen the soft
material (NaCl).
Kinsland and Bassett, 1977
Meade and Jeanloz, 1988
Funamori et al., 1994
Weidner et al., 1994
Pure NaCl (gradient)
MgO-NaCl (1:2)
MgO-NaCl (1:3)
Pure NaCl
2.5
Differential Stress (GPa)
2.0
1.5
1.0
0.5
0.0
0
10
20
30
Pressure (GPa)
40
50
Figure 6.7. Comparison of differential stress of NaCl B1 and B2 phases at elevated
pressures. Solid blue symbols are results from this experiment, MgO-NaCl (1:2)
mixture; Solid pink symbols (MgO-NaCl (1:3) are from chapter 5; Solid green
circles (pure NaCl) are from chapter 4. It clearly shows that the strength of NaCl
(with 33.3% MgO) is larger than that of NaCl (with 25% MgO) and pure NaCl,
indicating the strong material enhances the strength of the soft material. Open
symbols are from previous studies (Kinsland and Bassett, 1977; Meade and Jeanloz,
1988; Funamori et al., 1994; Weidner et al., 1994)
6.2.4 Elasticity Study of NaCl
Based on the theory introduced in chapter 3, elastic constants can be deduced from m 0
and m1. And m0 and m1 can be derived from the plots of Q (hkl) versus 3Γ, The plot of
81
elastic constants versus pressures is shown in Figure 6.8. It is clearly shown that large
discrepancy observed in B2 phase. The discrepancy may be caused by plastic
deformation when sample is compressed to exceed the yield point. To evaluate elastic
and plastic deformation, strain and stress state need to be considered.
400
Kiefer (unpublished)
Xiao et al.,2006
Whitefield et al.,1976
Bass et al.,2006
NaCl (1:2), this study
Elastic Constants (GPa)
300
200
C11 (NaCl B2)
C11 (NaCl B1)
C12 (NaCl B2)
100
C44 (NaCl B2)
C12 (NaCl B1)
C44 (NaCl B1)
0
0
4
8
12
16
20
24
28
32
Pressure (GPa)
36
40
44
48
52
Figure 6.8. Comparison of single crystal elastic constants of NaCl obtained from the
lattice strain theory, theoretical calculations and Brillion data. Solid symbols are
from this study. Open symbols, dashed and dash-dotted lines are from previous
published reports (Xiao et al., 2006; Bass et al., 2006; Whitefield et al., 1976)
6.2.5 Strain of NaCl
As pressure increases, sample undergoes a large amount of stresses, causing strain
developed inside the sample. The change of micro-strain and grain size can affect the full
width at half maximum (FWHM) of peaks. Using peak broadening method described in
chapter 3 on the basis of expression 3.36, B2 cos2θ as a function of sin2θ is plotted (Figure
6.9). The slope can provide information of micro-strain and the intercept can be used to
estimate grain size. The full widths at half maximum of 2-4 peaks at each pressure were
82
employed, showing different slopes with different intercepts. Generally, the slopes
increase as pressures increase for B1 and B2 phase respectively, indicating strain increase
as well. However, there is a decrement occurred at 43.1 GPa, suggesting a hardening
process.
NaCl
0.15
200
40.6 GPa
0.10
2
B cos θ
25.5GPa
2
38.8 GPa
110
0.05
34.5 GPa
43.1GPa
100
33.5 GPa
10.7 GPa
3.8GPa
0.00 111 200
6
222
220
8
10
12
sin θ
2
14
16
18
20x10
-3
Figure 6.9. Plot of sin2θ versus B2cos2θ variation for NaCl B1 and B2 phases at
different pressures. B: full width at half maximum. Miller indices are labeled with
data. The solid lines are linear fits to the experimental data. The slope of the straight
line reflects the grain size. Within B1 phase range, the slope increases and arrives to
highest value at 25.5 GPa; For B2 phase, the strain (slope) increase from 33.5 to 40.6
GPa, while a drop occurred at 43.1 GPa.
In Figure 6.10, NaCl B1 phase is represented by green symbols and line and B2
phase is shown by pink symbols and line. Pressures are also labeled. For NaCl B1 phase,
strains linearly increase with stresses suggesting an elastic deformation process. For B2
phase, strain-stress curve shows curve rather than linear line. A clear sign of plastic
deformation is already occurred. Therefore, elastic deformation may already occur before
30-33 GPa and then plastic deformation follows at high pressure for NaCl B2 phase.
83
1.2
25.5
40.6
43.1
1.0
38.8
Strain
34.5
33.5
0.8
NaCl B2
10.7
0.6
NaCl B1
0.4
3.8
0.2
0.4
0.6
0.8
1.0
1.2
Stress (GPa)
1.4
1.6
1.8
Figure 6.10. Variations of strain as a function of stress for NaCl B1 and B2 phases.
Green symbols and line are NaCl B1 phase, and purple symbols and line denote
NaCl B2 phase. Pressures are labeled with data point. In B1 phase, strain increases
linearly with stress, which suggests an elastic deformation process. For B2 phase,
strain increase non-linearly with stress from 33.5 to 43.1 GPa suggesting a plastic
deformation already developed.
6.2.6 Strain of MgO
Linear fits to the experimental data can be traced by plotting a graph with sin2θ versus B2
cos2θ. The slope yields information of micro-strain due to the grain-to-grain contact
under compression, and the intercept is related to inverse of the grain size (Figure 6.11a).
The full widths at half maximum of 2-4 peaks at each pressure were used in the plot,
showing different slopes with different intercepts. Generally, B2cos2θ values become
higher as pressures increase, suggesting the broadening of lines. However, a drop
occurred to the linear fit at 30.7 GPa, which is the phase transition boundary for NaCl B1
and B2.
Figure 6.11(b) shows the microscopic deviatoric strain distribution of MgO at
different pressures, which was derived from the slope of the lines in Figure 6.11(a).
Strains of MgO rise as pressures increase and can be fit as a linear line, which suggesting
84
an elastic deformation process. At higher pressure, i.e. more than 31 GPa, MgO 200
peaks are overlapped by Be peaks, their full widths at half maximum are not reliable to
deduce strain information. But, it can be concluded that MgO is subject elastic
deformation at least up to 25.5 GPa.
0.16
(a)
(b)
1.4
220
1.2
0.14
25.5 GPa
1.0
200
21.2 GPa
Strain
0.10
2
2
B cos θ
0.12
0.08
0.8
30.7 GPa
0.6
0.06
10.7 GPa
0.4
0.04
111
3.8 GPa
0.02
8
10
12
14
2
sin θ
16
18
0.2
20x10
-3
5
10
15
20
Pressure (GPa)
25
30
Figure 6.11. (a) Plot of sin2θ as a function of B2cos2θ for MgO at different pressure.
The FWHM and two theta values of MgO 111, 200 and 220 peaks were used for the
fittings. (b) The variations of strain as a function of pressure for MgO up to 25.5
GPa. The strain increases linearly with pressure, which suggests an elastic
deformation process.
6.3
Conclusions
Radial ADXD study on MgO-NaCl (1:2) mixture was investigated at pressure up to 43.1
GPa at room temperature. The d-spacings varied with different ψ angles. The highest dspacings were found at ψ=90º, indicating the smallest stress condition. In this experiment,
phase transition from B1 to B2 started at 30.7 GPa, and completed at around 38.8 GPa,
which is slightly higher than those of pure NaCl and NaCl in the MgO-NaCl (1:3)
mixture. The involvement of strong materials (MgO) may extend NaCl B1 phase to high
pressure. The strength of NaCl B1 111 are higher than that of NaCl B1 200, suggesting
that B1 111 will lastly yield and B2 200 will firstly deform under high pressure. The
85
differential stresses of NaCl mixed with 33.3% MgO are higher than that of pure NaCl,
and NaCl mixed with 25% MgO, indicating that the strong material MgO enhances the
strength of the soft material NaCl. Based on peak broadening study, NaCl B1 was within
elastic deformation, however, NaCl B2 phase was subject to plastic deformation before
33 GPa. In the case of MgO, it was under elastic deformation at least up to 25.5 GPa.
86
Chapter 7
7
Strength and Strain Study on MgO-NaCl (1:1) Mixture
In this chapter, MgO-NaCl mixture with a ratio of 50 vol% of MgO and 50 vol% of NaCl
was studied in a symmetric DAC by ADXD at X17C, NSLS, Brookhaven National
Laboratory. The primary goals are (1) to compare the strength of NaCl obtained from
chapter 4, 5 and 6; (2) to explore how the phase boundary between NaCl B1 and B2 are
affected by different volume of MgO; (2) to study the stress and strain curves of NaCl; (3)
to obtain the stress and strain of MgO; (4) to construct the strength model with different
ratios of the MgO-NaCl mixture.
7.1.1 Experiments
Two experimental runs with MgO-NaCl (1:1) mixture were performed in this work. The
first run was performed at pressure up to 31.6 GPa, and the second run was conducted at
pressure up to 44.4 GPa. The reason for the second experiment is to obtain data of NaCl
B2 phase, which generally is observed at pressure greater than 30GPa. Same starting
materials were used in both experiments. NaCl powder (99.999%purity) and MgO
powder (99.999% purity) from Alfa Aesar were weighed by Mettler Toledo balance into
1:1 volume ratio. The mixture was mechanically ground down to a few microns
examined under microscope. Acetone was employed during grinding to reduce the heat
produced and to ensure the homogeneity of mixture of two powder samples. Due to the
MgO and NaCl mixture exposures to air, the moisture may be retained with the mixed
sample. The mixed sample was heated in a furnace to dry out at a temperature of 120 °C
for overnight.
In the first experiment, a symmetric DAC with a pair of 300 μm culet size
diamonds were used as pressure apparatus. An X-ray transparent beryllium gasket with a
120 μm hole was used as sample chamber. A thin gold foil was placed at the center of the
anvil served as pressure calibrant. Pressure was determined by the EOS of Au (Fei et al.,
2007). Radial ADXD measurement was carried out at X17C, NSLS. The X-ray
wavelength was fixed at 0.4075 Å. A charge-coupled device (CCD) area detector was
87
employed to acquire diffraction data. Note that due to the loss of the gold diffraction
patterns at pressure greater than 31.6 GPa in the first run, the experiment was stopped.
In the second experiment, the mixture was loaded into a 100 μm diameter sample
chamber in a beryllium gasket. A gold piece was placed at the center of the anvil, serving
as pressure standard. The sample was compressed by a pair of 300 μm diamonds in a
symmetric DAC. The DAC was placed on the experiment stage in a radial geometry. The
X-ray beam was directed through the beryllium gasket to the sample. ADXD
measurement was conducted. The X-ray wavelength was fixed at 0.4066 Å. The pressure
was carried up to 44.4 GPa, which NaCl B2 phase can be observed and analyzed. 2-D Xray diffraction images were obtained for a collection time of 15 minutes. To minimize the
uncertainty of the pressure determination during each compression step, the sample was
allowed to relax for at least 30 minutes for stress relaxation. Then the Fit2D program was
employed to deduce the 2D images into 1-D patterns set a five-degree interval.
7.2 Results and Discussions
7.2.1 1-D Diffraction Patterns
MgO-NaCl (1:1) mixture was gradually compressed to 31.6 and 43 GPa, with a 3-4 GPa
and 2-3 GPa step, respectively. The 1-D patterns with different ψ angles were stacked
together and shown in Figure 7.1a. The diffraction patterns are presented from ψ=0 ° to
ψ=180 ° at the top. The Miller indices are also labeled with the patterns. The peaks
labeled with “Be” are contributions from beryllium gasket. It can be clear seen that at
different ψ angles, two theta values of the sample varied but not for Be peaks. Moreover,
it can be clearly observed that MgO 200, Au 200 and NaCl B1 220 show strong curvelike patterns compared to the unchanged Be peaks in Figure 7.1b. And the smallest two
theta value is found at around ψ=90°. The dashed lines in Fig 7.1b indicate the curve
tendency. Based on the Bragg law 2dsinθ=λ, two theta value has an inverse relation with
d-spacing. Therefore, small two theta values represent large d-spacings. On the other
hand, at ψ=90°, the samples have largest d-spacings and therefore under the minimum
stress condition. It can also be inferred that the sample would be under the largest stress
at both 0° and 180°.
88
2000
1000
Au 220
MgO 220
B1 222
B1 220
Au 200
MgO 200
BeO
Au 111
MgO 111
B1 111
BeO
1500
B1 200
6.7 GPa
(a)
Intensity
180º
500
90º
0
-500
0º
8.0
9.0
10.0
11.0
12.0
Two Theta
13.0
14.0
15.0
16.0
(b)
Figure 7.1. Diffraction patterns of MgO-NaCl mixture (1:1) collected at 6.8 GPa
were stacking together in a 5-degree angle step. (a) Miller indices are labeled with
peaks. B1 denotes NaCl B1 phase, BeO denotes beryllium Oxide, Au denotes gold. ψ
angle from 0° to 180° is labeled with axis . (b) The zoom-in plot with two theta at
10.5-12.50 shows that smallest two theta value appears at ψ=90° and largest values
at both 0° and 180°. Dashed line shows two theta changing tendency.
Each 2-D image was deduced into 1-D patterns at every five-degree angle interval.
Therefore, at each pressure, there are 37 diffraction patterns ranging from 0º to 180º.The
89
diffraction pattern at ψ=54.7º is regarded as at hydrostatic condition on the basis of
equation 3.4 (chapter 3). Hence, diffraction patterns at ψ=55º are regarded close to
hydrostatic condition. Figure 7.2 shows those diffraction patterns at ψ=55º under
different pressures. Miller indices of different phase are shown, together with pressures.
It can be clearly observed that all sample phases (except beryllium peaks) shift to higher
two theta direction with increasing pressure.

Be
MgO 220
B1 220
Au 220
31.6
MgO 200
BeO
BeO
B1 111
4000
Intensity
B1 200
5000
Be
Au 111
6000
3000
20.7
17.7
0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
Two Theta
Au 220
MgO 220
12.3
Be
Be
B1 222
B1 220
Au 200
Be
BeO
MgO 200
Au 111
B1 111
B1 200
MgO 111
2000
1000
23.6
6.7
14.0
15.0
16.0
17.0
Figure 7.2. Diffraction patterns of MgO-NaCl (1:1) mixture collected at hydrostatic
condition up to 31.6 GPa. B1 denotes NaCl B1 phase and Au denotes gold. Pressures
are labeled with each pattern.
7.2.2 Phase Transformation
As pressure increases, sample peaks shift to higher two theta (i.e. small d-spacing)
direction, indicating that sample is subject to compression. The volumes of sample
become smaller in response to the new environment. Eventually phase transition occurred
at transition pressur. As shown in Figure 7.3, NaCl B2 110 peak first appeared at 33.1
GPa, accompanied by the fade out of NaCl B1 peaks. However, B1 200, the most intense
peak for B1 phase was still observable even at the highest pressure.
90
In addition to the peak shiftings, peak broadenings occurred simutaneously while
pressure increase. Peak broadening features can be easily observed in B2 110. At about
33.1 GPa, B2 110 shows sharp and narrow feature. As pressure increase, B2 110 peaks
exhibit broad feature to reflect the stress condition within the sample, similar as other B2
phase peaks. It is known that the grain size and micro-strain inside the sample can affect
peak shapes. Therefore, peak broadening can be used to evaluate grain size and microstrain. More details will be discussed in later section.
Figure 7.3 Diffraction patterns of MgO-NaCl (1:1) mixture collected at hydrostatic
condition up to 44.4 GPa. The arrows show the emergence of NaCl B2 phase started
at 33.1 GPa and accomplished fully at 38.8 GPa. B1 denotes NaCl B1 phase, B2
denotes NaCl B2 phase, Au denotes gold and Be denotes beryllium. Pressure is
labeled with each pattern.
It has been well established that NaCl with a rock salt structure (the B1 phase)
transforms into a CsCl structure (the B2 phase) at about 30 GPa and room temperature
91
(Sata et al., 2002; Yagi et al., 1983; Bassett et al., 1968; Sato-Sorenson, 1983; Heinz and
Jeanloz, 1984; Nishiyama et al., 2003; Xiao et al., 2006; Ono, 2008). However, most of
the studies focus on the starting point of transition pressure, and rarely report the pressure
at which phase transformation completed. Moreover, phase transformation of NaCl in a
composite environment may behavior differently from that pure NaCl. The phase
transition and phase boundary of NaCl B1 and B2 in a mixture condition hasn’t been well
established yet. My results showed that phase transition boundary between NaCl B1 and
B2 phase are slightly varied with different MgO-NaCl mixture to pure NaCl. In this study,
the transition started at 33.1 GPa and B1 remains stable to the maximum applied pressure
of 44.4 GPa while for the pure NaCl it was found to transform to B2 at 29.4 GPa and
completed at 31.7 GPa. For MgO-NaCl (1:3) mixture the B1-B2 transition started at 29.8
GPa and finished at 36.7 GPa. For MgO-NaCl (1:2) mixture the B1-B2 transition began
at 30.7 and completed at 38.8 GPa (Table 7.1).
Table 7-1 Comparison of NaCl B1 and B2 phase transition in different studies
Author
Year
Composition
Pressure
(GPa) at
start of
transition
Bassett et
al.
1968
NaCl
30 GPa
XRD (DAC)
Yagi et al.
1983
NaCl
27 GPa
XRD (DAC)
SatoSorensen
1983
NaCl
29 GPa
XRD (DAC)
Heinz and
Jeanloz
1984
NaCl
29 GPa
XRD (DAC)
Li and
Jeanloz
1987
NaCl
30 GPa
Visual (DAC)
Hofmeister
1997
NaCl (in petroleum
jelly)
32 GPa
IR (DAC)
Sata et al.
Nishiyama
et al.
2002
NaCl :(MgO+Pt) 2:1
32.08 GPa
2003
NaCl -MgO (1:1)
29.3 GPa
XRD (DAC)
XRD(MultiAnvil)
Completion
Techniques
92
This study
NaCl
MgO-NaCl (1:3)
29.8 GPa
29.4 GPa
32.3 GPa
36.7 GPa
XRD (DAC)
XRD (DAC)
MgO-NaCl (1:2)
30.7 GPa
38.8 GPa
XRD (DAC)
MgO-NaCl (1:1)
33.1 GPa
44.4 GPa
XRD (DAC)
My studies show a trend that the phase boundary between NaCl B1 and B2 are
varied with the volume friction of MgO. In Figure.7.4, the plot of the phase transition
pressure versus NaCl and MgO composition shows that the more MgO added to NaCl
mixture, the higher transition pressure is observed and the two-phase co-existing regime
is also extended. The B1-B2 transition boundary for MgO-NaCl binary system can be
expressed as P=0.063061 x+29.043, where P stands for pressure and x is the volume ratio
for MgO. For completion of the NaCl B1 phase, it can be expressed as P=0.23718
x+31.628. There is very little report on the study of disappearance of NaCl B1 phase,
especially for NaCl in a mixture environment. Therefore, Figure 7.4 suggests the
appearance of MgO strongly affects the two-phase existing zone and completion of the
transition but slightly extends the B1-B2 transition pressure of NaCl.
Figure 7.4 B1-B2 transition pressure for MgO-NaCl binary system. Solid green
symbols represent the emergence of B2 phase and solid pink circles represent the
completion of B1-B2 phase transition.
93
7.2.3 Strength of NaCl
Reliable two theta values and full width at half maximum (FWHM) of the mixed MgONaCl sample were obtained by PeakFit program. Figure 7.5 shows d-spacings of NaCl
B1 111 and MgO 200 as a function of ψ angle. The pressure, labeled with each pattern,
also shown, from 6.7 to 23.6 GPa. The d-spacings are generally decreased with pressure
increasing. Moreover, the d-spacings also varied with ψ angle at same pressure step,
shown by a sinusoid function. In general, the maximum d-spacing is found at ψ=90° and
the minimum d-spacing values found at 0° and 180° , indicating that the sample
undergoes minimum stress at 90°.
2.08
3.10
6.7
6.7
2.07
d_spacing (angstrom)
B1 111
MgO 200
3.05
2.06
2.05
3.00
12.3
12.3
2.04
2.95
17.7
2.03
17.7
2.90
20.7
2.02
20.7
23.6
2.85
0
25
50
75
100
125
Azimuthal Angle ψ
150
175
23.6
2.01
0
25
50
75
100
125
Azimuthal Angle ψ
150
175
Figure 7.5 Plot of NaCl B1 110 and MgO 200 d-spacings versus ψ angles show
sinusoid curves from 6.7 to 23.6 GPa. Pressure (hydrostatic condition) is labeled
with each curve. The d-spacings are decreasing with pressure. The largest d-spacing
shows at around 90° for all pressures.
According to equation 3.4 (chapter 3), the observed d-spacings fit linearly with 13cos2 ψ. The intercepts yield hydrostatic d-spacings, from which hydrostatic pressure can
be derived. The slopes reveal information of Q(hkl), the ratio of differential stress over
shear modulus. With input of shear modulus, the differential stress can be obtained. In
Figure 7.6, the typical d-spacing as a function of (1-3cos2 ψ) for NaCl B1 111 and MgO
200 are presented. Linear functions were clearly displayed. The pressure range is same as
94
Figure 7.5. If all d-spacing values well fit with a smooth and symmetric sinusoid curve in
Figure 7.5, the consequent d-spacing versus (1-3cos2ψ) plot will show a perfect linear
relationship. The non-linearity in the dm(hkl) versus (1-3cos2ψ) plots is attributed to the
shift of the sample off from the center of the anvil face, lack of parallelism of the anvil
faces, and the presence of texture in the sample (Singh, 2000).
6.7
6.7
2.08
3.10
2.07
MgO 200
3.05
d_spacing (angstrom)
d_spacing (angstrom)
B1 111
2.06
12.3
12.3
3.00
2.95
17.7
20.7
2.04
23.6
17.7
20.7
2.90
-2.0
2.05
23.6
-1.5
-1.0
-0.5 2
1-3cos ψ
0.0
0.5
1.0
2.03
2.02
-2.0
-1.5
-1.0
-0.5 2
1-3cos ψ
0.0
0.5
1.0
Figure 7.6 NaCl B1 111 and MgO 200 d-spacings as a function of 1-3cos2 ψ at
pressures of 6.7-23.6 GPa. Linear fits can be clearly observed. Pressure is labeled
with each linear fit. The d-spacings decrease as pressures increase. Dashed lines
indicate the intercept and also the hydrostatic condition.
The ratio of differential stress to shear modulus, t/G, is directly obtained from the
slopes of the d-spacing as function of (1-3cos2ψ) with assumed Reuss conditions in the
sample. This is directly from experimental data with no contribution from any theoretical
input. Figure 7.7 shows that the ratios of differential stress over shear modulus of B1 111
are much larger than those of B1 200, which means d-spacing of B1 111 changed faster
with ψ angle than that of B1 200. A strength weakening occurred along the phase
transition from B1 to B2. For B2 phase, differential stresses over shear modulus of
different planes show similar increment.
95
Differential Stress/Shear Modulus
4.5x10
MgO-NaCl (1:1)
-3
B1 111
4.0
B2 200
3.5
B2 100
B1 220
3.0
B2 110
2.5
B1 200
2.0
8
12
16
20
24
28
32
Pressure (GPa)
36
40
44
Figure 7.7 Plots of differential stresses over shear modulus as a function of pressures
for MgO-NaCl (1:1) mixture. B1 denotes NaCl B1 phase and B2 denotes NaCl B2
phase. B1 111 shows the highest values for B1 phase. For B2 phase B2 200 shows
the highest value.
With the input of shear modulus, the differential stress can be obtained according
to equation 3.6 (chapter 3) assuming the samples are under iso-stress conditions. The
NaCl shear modulus value is obtained from Brillouin scattering data (Bass, 2006;
Lakshtanov, 2007). In Figure 7.7, the differential stresses of MgO-NaCl (1:1) mixture are
compared to the strength of pure NaCl, MgO-NaCl (1:3) mixture, MgO-NaCl (1:2)
mixture and previous published reports (Meade and Jeanloz, 1988; Funamori, 1994;
Kinsland and Bassett, 1977; Weidner, 1994). NaCl B1 phase in MgO-NaCl (1:1) mixture
shows the differential stress from 0.38(6) to 1.25(6) GPa in the pressure range of 6.7–
23.6 GPa. A strength softening occurred around the B1-B2 phase transformation. The
differential stress of NaCl B2 phase of MgO-NaCl (1:2) mixture increases to 2.03(6) GPa
at the maximum applied pressure of 44.4 GPa. In this experiment, the differential stress
and pressure can be expressed by (1) for NaCl B1 phase: t=0.048937P - 0.017855; (2) for
NaCl B2 phase: t=0.125850P- 3.522382. Obviously, the t values of NaCl in composite
environment (mixed with MgO at 1:1 volume ratio) are much larger than that of pure
96
NaCl, NaCl mixed with 25% MgO and 33.3% MgO (see chapter 4, 5 and 6). In general,
the strength of NaCl B1 in 1:1 mixture environment is about 15-280% larger than pure
NaCl, 5-130% larger than NaCl in MgO-NaCl (1:3) mixture condition, and 5-65% higher
than NaCl in MgO-NaCl (1:2) mixture condition. The strength of NaCl B2 phase in
MgO-NaCl (1:1) mixture environment is 145-240% higher than that that of pure NaCl,
67.5-97% larger than that of NaCl B2 in MgO-NaCl (1:3), and 27.5-34% larger than that
of NaCl B2 in MgO-NaCl (1:2) mixture. It is therefore concluded that the strong material
(MgO) can strengthen the soft material (NaCl).
Kinsland and Bassett, 1977
Meade and Jeanloz, 1988
Funamori et al., 1994
Weidner et al., 1994
Pure NaCl (gradient)
MgO-NaCl (1:1)
MgO-NaCl (1:2)
MgO-NaCl (1:3)
Pure NaCl
2.5
Differential Stress (GPa)
2.0
1.5
1.0
0.5
0.0
0
10
20
30
Pressure (GPa)
40
50
Figure 7.8 Comparison of differential stresses of NaCl B1 and B2 phases at elevated
pressures with previous studies (Kinsland and Bassett, 1977; Meade and Jeanloz,
1988; Weidner et al., 1994; Funamori et al., 1994). Solid orange symbols are results
from this experiment, MgO-NaCl (1:1) mixture; Solid blue symbols MgO-NaCl (1:2)
are from chapter 6; Solid pink circles (MgO-NaCl (1:3)) are from chapter 5; Solid
green circles (pure NaCl) are from chapter 4. It clearly shows that the strength of
NaCl (with 50% MgO) is larger than that of NaCl (with 33.3% MgO), NaCl (with 25%
97
MgO) and pure NaCl, indicating the strong material enhances the strength of the
soft material.
7.2.4 Strain of NaCl
As pressure increases, sample suffers higher stresses, causing strain developed inside the
sample. The change of micro-strain and grain size can affect the full width at half
maximum (FWHM) of sample peaks. Based on peak broadening theory, sin2θ versus B2
cos2θ plot shows that the slope can provide information of micro-strain and the intercept
can be used to estimate grain size (Figure. 7.9). The full widths at half maximum of 2-3
peaks of NaCl B1 and B2 phase at each pressure were taken into consideration. Generally,
the slopes increase as pressures increase, indicating strain increase as well. However,
there is a decrement occurred at 42.3 GPa, suggesting a hardening process.
0.20
NaCl B2
200
44.4 GPa
0.15
2
B Cos 
42.3 GPa
0.10
39.5 GPa
2
110
35.4 GPa
0.05
100
34.8 GPa
6
8
10
12
2
Sin 
14
16
18x10
-3
Figure 7.9 Plot of sin2θ versus B2cos2θ variation for NaCl B2 phases at different
pressures. The strain (slope) increase from 34.8 to 39.5 GPa, while a drop occurs at
42.3 GPa.
In Figure 7.10, NaCl B1 phase is represented by green symbols and line and B2
phases is shown by pink symbols and line. Pressures are also labeled with each data point.
For NaCl B1 phase, strain goes up as stress increase and can be fit as a linear line to
about 17.7 GPa which suggests an elastic deformation limit. For B2 phase, strain increase
98
with a gentle slope as stress increasing, no longer a straight line. A clear sign of plastic
deformation already occurred. Therefore, elastic deformation may already occur before
20.7 GPa and then plastic deformation follows at high pressures including NaCl B2 phase.
1.6
42.3
39.5
1.4
B1
Strain
1.2
6.7
35.4
44.4
B2
17.7
1.0
0.8
0.6
34.8
6.7
0.4
0.6
0.8
1.0
1.2
1.4
Stress (GPa)
1.6
1.8
2.0
Figure 7.10 Variations of strain as a function of stress for NaCl in this study. Green
circles and line are NaCl B1 phase whereas pink symbols and line denote NaCl B2
phase. Pressures are labeled with data point. In B1 phase, strain increases linearly
with stress from 39.5 to 44.4 GPa suggesting a plastic deformation already
developed.
7.2.5 Strength and Strain Study of MgO
The differential stress of individual phase in an MgO-NaCl (1:1) mixture can be obtained
successfully, including MgO. To retrieve the strength of MgO, the expression 3.6 and
highly precise shear moduli from Brillouin studies (Zha et al., 2000) for MgO was
employed. Figure 7.11 shows the differential stress supported by MgO obtained in this
study compared to the results from previous measurements.
It is noted that the results of this study lie in the middle compared to those
reported previously from lattice strain theory (Kinsland and Bassett, 1977; Duffy et al.,
99
1995; Uchida et al., 2004; Merkel et al., 2002; Akhmetov, 2008), Brillioun scattering
(Gleason et al., 2011), pressure gradient method (Meade and Jeanloz, 1988a), and
diffraction line width analysis (Singh et al., 2004). The strength of MgO in this
experiment is lower than that reported by Singh (2004), Merkel (2002), Kinsland and
Bassett (1977), Duffy (1995) and Uchida (2004); and higher than those studied by Meade
and Jeanloz (1988), Gleason (2011) and Akhmetov (2008).
Singh (2004) studied MgO powder of high purity and with an average crystalline
size of 60 nm. He concluded that very fine grain sizes (comparing to a grain size in my
experiment which were around 3 μm) could lower the porosity and defect and therefore
strengthen MgO, with a high shear stress of ~10 GPa at a pressure of 55 GPa. Pure MgO
without mixing any other material showed plastic deformation appears at very lower
pressure (<10 GPa) (Weidner et al.2004). Merkel et al. (2002) studied fine grain size
MgO (less than 1 μm) and placed a small pallet of Fe (served as pressure scale) in
between two pallets of MgO sample. They reported strength of MgO reached about 8
GPa at the applied pressure of 10 GPa. The sample MgO is believed to transform to
plastic deformation at higher applied pressure.
In comparison with the strength data of MgO from Merkel et al. (2000), my
results are lower than theirs. That could be because (1) small percentage of MgO sample
used in this study; (2) slightly large grain sizes at ~3 um; (3) the strength effect arising
from contribution of Fe used in Merkel et al. (2000) remains unclear. Uchida et al. (2004)
investigated pure MgO, mixed with boron and epoxy. Due to the high strength property
of boron, it likely enhances the strength of MgO, compared to NaCl in this case. Duffy
(1995) utilized pure MgO and 5wt% molybdenum as test material. Similarly, it is inferred
that “strong” molybdenum may also contribute to the strength result. Kinsland and
Bassett (1977) examined mechanical properties of MgO in MgO-NaCl mixture; it also
showed higher values of the supported differential stress by MgO than the results in this
study. However, the authors did not report the volume ratio of individual phases in the
studied MgO and NaCl mixture, which is very critical (e.g. Li et al., 2007). Based on the
dataset provided by this thesis, it is inferred that Kinsland and Bassett (1977) examined
the MgO-NaCl mixture may have more than 50 vol% of MgO. Gleason et al. (2011)
100
filled 4:1 methanol and ethanol as pressure medium into the sample chamber with MgO.
This created a quasi-hydrostatic environment, reducing the supported stress for MgO.
That is the reason their data are lower than data in this experiment. Meade and Jeanloz
(1988) studied pure MgO by pressure gradient method and this method is under debate (e.
g. Jing et al., 2007). Therefore, the method may cause the lower shear stress values than
my data. Akhmetov (2008) reported strength of MgO in MgO and NaCl (1:4) mixture,
the MgO is immersed into the NaCl. It seems that high NaCl volumes act like effectively
reducing the shear stress development in the sample and there is no plastic deformation in
his study. .
It is concluded that there are several factors to affect the strength behavior of
MgO: (1) the friction of MgO: the higher volume ratio of MgO, the higher differential
stress observed; (2) crystallite size: the smaller grain size will result in less porosity and
defect and higher strength; (3) strain percentage: Uchida and colleagues suggested that
any “differential stress” level at 5 - 10% strain should be flow stress. Flow stress of MgO
depends strongly on total strain, due to strain hardening; (4) pressure medium: soft
pressure mediums will minimize the difference between σ1 and σ3, which is differential
stress.
Soft material is demonstrated to affect the strength of a strong material in a mixture
environment. Weak phase by forming a matrix surrounding the strong phase,
accommodating most of the strain caused by strong material during compression.
Importance of the weaker phase for the rheology of a multiphase material was reported
from previous deformation studies (Bloomfield and Covey-Crump, 1993; Jordan, 1998).
Bloomfield and Covery-Crumpy studied mechanical properties of calcite (stronger phase)
and halite (weaker phase) in a mixture and found that a large fraction of strain is
partitioned in weaker halite. Jordan confirmed the rheology of well foliated rock is
dominated by the rheology of the weakest phase. In my study, it seems that strong
material could strengthen soft material. On the other hand, strong material can be
softened by weak material as well. However, it is very hard to define strain rate in the
diamond anvil cell and therefore, it is very hard to directly compare previously obtained
results using different techniques.
101
The differential stress of MgO phase increase linearly with pressure up to 4.3 GPa at
the pressure of 23.6 GPa. Differential stress of MgO in the MgO-NaCl (1:1) mixture at
6.7-23.6 GPa can be described by t=0.164041P- 0.084616, where t and P are the
differential stress and the pressure in GPa. By studying differential stress, the lower
bound of yield strength information could be attained.
12
MgO 200
MgO 220 (Kinsland & Bassett, 1977)
small strain
large strain (Meade&Jeanloz, 1988)
Duffy et al.,1995
run 1
run 2 (Merkel et al.,2002)
Uchida et al., 2004
Singh et al., 2004
Akhmetov, 2008
Gleason et al.,2011
MgO (1:1), this study
Differential Stress (GPa)
10
8
6
4
2
0
0
10
20
30
Pressure (GPa)
40
50
60
Figure 7.11 Comparison of differential stresses of MgO as a function of pressure
obtained from this study with previous data (Kinsland and Bassett, 1977; Meade
and Jeanloz, 1988; Duffy et al., 1995; Merkel et al., 2002; Uchida et al., 2004; Singh
et al., 2004; Akhmetov, 2008; Gleason et al., 2011). Solid red circles: this study; red
line: power fit for Singh, 2004; green open diamonds: run1 and run 2 in Merkel,
2002; green line: linear fit from Duffy, 1995; black line: linear fit from Meade and
Jeanloz, 1988, large strain; blue line: linear fit for Akhmetov, 2008; purple and
black circle: Kinsland and Bassett, 1977; blue dashed lines: Uchida et al., 2004
102
To evaluate whether MgO underwent elastic or plastic deformation, strain
measurements are discussed here. As pressure increases, sample undergoes a large
amount of stresses, causing strain developed inside the sample. The change of microstrain and grain size can affect the full width at half maximum (FWHM) of peaks. Using
peak broadening method, plot of sin2θ versus B2 cos2θ shows the slope can provide
information of micro-strain and the intercept can be used to estimate grain size (Figure
7.12). The full widths at half maximum of 2-3 peaks for MgO phase at each pressure
were taken into consideration. It is clear seen that the slopes increase as pressures
increasing in the pressure range of 6.7-23.6 GPa, indicating strain increase as well. At
higher pressure, i.e. 44.4 GPa, MgO 200 peak are overlapped with beryllium peak.
Therefore, the full width at half maximum of MgO 200 is not reliable to plot Bcos2θ
versus sin2θ graph. Furthermore, the strain of MgO at higher pressure was not observable
in this experiment.
0.16
MgO
23.6 GPa
0.14
0.12
20.8 GPa
2
2
B Cos 
0.10
0.08
17.7 GPa
0.06
12.3 GPa
0.04
6.7 GPa
220
0.02
111
200
0.00
8
10
12
14
2
Sin 
16
18
20x10
-3
Figure 7.12 Plot of sin2θ versus B2cos2θ variation for MgO phases at from 6.7 to 23.6
GPa. Miller indices are labeled with data. The slopes become steeper with pressure
increase.
Figure 7.13 shows the strain of MgO plotted as a function of stress. Pressures are
also labeled with every data point. During the whole pressure range from 6.7-23.6 GPa,
103
strain of MgO goes up linearly with stress increase and seems can be regarded as an
elastic deformation. Although the strain values seem high, MgO may have not reached
plastic deformation yet with the buffer of NaCl.
1.3
MgO
23.6 GPa
1.2
Strain
1.1
1.0
17.7 GPa
20.7 GPa
0.9
0.8
0.7
6.7 GPa
1.5
12.3 GPa
2.0
2.5
3.0
Stress (GPa)
3.5
4.0
Figure 7.13 The variations of strain as a function of stress for MgO in this study.
Pressures are labeled with data point. The strain increases linearly with stress,
which suggests an elastic deformation for MgO at pressure range of 6.7-23.6 GPa.
7.3 Conclusions
Radial ADXD study on MgO-NaCl (1:1) mixture was investigated at pressure up to 44.4
GPa at room temperature. The d-spacings varied with different ψ angles and the highest
d-spacings were found at ψ=90º, indicating the smallest stress condition. In this
experiment, phase transition from B1 to B2 started at 33.1 GPa, and B1 remains
observable to the maximum pressure of 44.4 GPa, which is higher than those of pure
NaCl, NaCl in the MgO-NaCl (1:2) and (1:3) mixture. The involvement of strong
materials (MgO) may extend the stability of NaCl B1 phase to much higher pressure. The
strength NaCl B1 111 are higher than other NaCl B1 planes, suggesting that B1 111 will
yield last and B2 200 will deform first under high pressure. The strength of NaCl with 50%
MgO are higher than that of pure NaCl, NaCl mixed with 25% MgO and NaCl mixed
with 33.3% MgO, indicating that the strong material MgO enhances the strength of the
104
soft material NaCl. Based on line broadening study, NaCl B1 shows plastic deformation
after 17.7 GPa, and NaCl B2 phase was subject to plastic deformation after ~30 GPa.
MgO was under elastic deformation at least up to 23.6 GPa. The stresses supported by
MgO mixed with NaCl are lower than that pure MgO experiments (Sing et al., 2004;
Merkel et al., 2002; Duffy et al., 1995). It can be concluded that strong material could
strengthen the “soft” material; conversely “soft” material may also have influence on the
strong material in a mixture.
105
Chapter 8
8
Conclusions
In situ high pressure radial X-ray diffraction studies using diamond anvil cell were
performed at beam line X17C, NSLS, Brookhaven National Laboratory. Four different
starting materials, pure NaCl, MgO-NaCl (1:3), MgO-NaCl (1:2) and MgO-NaCl (1:1)
mixture, were investigated in this study. Pure NaCl was studied by radial energy X-ray
diffraction (EDXD) at pressure up to 43.7 GPa. Radial ADXD experiments using MgONaCl (1:3), (1:2) and l (1:1) mixture were conducted at pressure to 57.6, 43.1 and 44.4
GPa, respectively.
In the experiment of pure NaCl, phase transformation from B1 (rock-salt structure)
to B2 (CsCl structure) started at 29.8 GPa, and completed at around 32.3 GPa. The
differential stresses of B1 phase increased from 0.22(6) to 0.38(6) GPa at pressure range
of 10.2 - 29.8 GPa, indicating that the differential stress of NaCl B1 phase underwent
small changes under compression. The differential stress supported by NaCl B2 phase
increased remarkably and reached 0.8(6) GPa at the maximum applied pressure of 43.68
GPa. The differential stress, t, as a function of pressure of B1 and B2 phase can be
expressed by t=0.009156P + 0.118155 and t=0.055198P -1.636567, respectively, where t
and P are both in GPa. The differential stress of NaCl B1 showed very gentle increment
with pressure, which suggested that NaCl B1 is a very good pressure-transmitting
medium at pressure below 30 GPa. However, the differential stresses supported by NaCl
became abruptly increase for B2 phase and no longer regarded as “soft” pressure medium
at higher pressures. Based on peak broadening study, the deformation of NaCl B1
remained in elastic regime, whereas B2 phase underwent a plastic deformation instead.
The elastic constants of B1 phase were calculated by lattice strain theory showed
reasonably agreement with previous published reports to 27 GPa, as above 27 GPa NaCl
was subject to plastic deformation. Differential stresses supported by different planes
showed that B1 200 had the lowest value, suggesting B1 200 may be responsible for the
initial deformation under pressure.
106
In the mixture of MgO-NaCl (1:3), NaCl B1 and B2 phase transformation occurred
at ~29.4 GPa and finished at around 36.7 GPa. The differential stresses of B1 phase
increased from 0.23(6) to 0.61(6) GPa at pressure of 5.2 to 28 GPa. The relation between
differential stress (t) and pressure can be described by t = 0.014751P + 0.210062. The
differential stress supported by B2 phase increased to 2.36(4) GPa at the maximum
applied pressure of 57.6 GPa. The differential stress – pressure relation of B2 phase can
be expressed as t = 0.078432P – 2.2658. The differential stresses of NaCl in a composite
environment exhibited higher values than those of pure NaCl. They are 65-75% higher
than pure NaCl B1 phase and 45-70% higher than pure NaCl B2 phase.
For MgO-NaCl (1:2) mixture, NaCl B1 and B2 phase transformation started at ~30.7
GPa and completed at around 38.8 GPa. The differential stresses of B1 phase increased
from 0.26(4) to 0.9(3) GPa at pressure of 3.8- 30.7 GPa. The relation between differential
stress, t, and pressure can be described by: t = 0.023923P + 0.166592, where t and P in
GPa. The differential stress supported by B2 phase increased to 1.52 GPa at the
maximum applied pressure of 43.1 GPa. The relation between differential stresses and
pressures can be expressed as t = 0.09959P – 2.810276. The stresses supported by NaCl
B1 phase in 1:2 mixture environment are about 50-120% higher than those of pure NaCl
and 5-35% higher than those of NaCl in MgO-NaCl (1:3) mixture. While in the case of
B2 phase, the obtained values are 90-160% higher than those of pure NaCl, and 30-50%
higher than those of NaCl B2 in MgO-NaCl (1:3) mixture. The strain-stress curve
suggested that NaCl B1 was subjected to elastic deformation up to 25.5 GPa, and B2 was
under plastic deformation after ~30 GPa. In the case of MgO, elastic deformation was
found at least up to 25.5 GPa.
In the experiment of MgO-NaCl (1:1) composite, phase transition from B1 to B2
started at ~33.1 GPa, and B1 remained observable to the maximum pressure of 44.4 GPa.
The differential stresses of B1 phase increased from 0.38(4) to 1.25(3) GPa at pressure of
6.7 - 23.6 GPa. The relation between differential stresses and pressures can be described
by: t = 0.048937 P - 0.017855. The differential stress supported by B2 phase increased to
2.02 GPa at the maximum applied pressure 44.4 GPa. Its differential stress and pressure
related in: t= 0.125850 P - 3.522382. In general, the stresses of NaCl B1 in MgO-NaCl
107
(1:1) mixture environment are about 15-280% higher than those of pure NaCl, 5-130%
higher than those of NaCl in MgO-NaCl (1:3) mixture, and 5-65% higher than those of
NaCl in MgO-NaCl (1:2) mixture. The stresses of NaCl B2 phase in MgO-NaCl (1:1)
mixture environment are 145-240% higher than those of pure NaCl, 67.5-97% higher
than those of NaCl B2 in MgO-NaCl (1:3) mixture, and 27.5-34% higher than those of
NaCl B2 in MgO-NaCl (1:2) mixture. The strain-stress study for NaCl B1 suggested
elastic deformation limited to 17.7 GPa, whereas B2 was under plastic deformation. In
the case of MgO, it was under elastic deformation at pressure up to 23.6 GPa. The
differential stress of MgO phase increased linearly with pressure up to 4.3 GPa at the
pressure of 23.6 GPa. Differential stresses of MgO in the MgO-NaCl (1:1) mixture at 6.723.6 GPa can be described by t = 0.164041P - 0.084616.
The stresses supported by NaCl in an MgO-NaCl mixture environment are larger
than that of pure NaCl. Conversely, the stresses supported by MgO in the MgO-NaCl
mixture are lower than that of pure MgO experiments (i.e. Sing et al., 2004; Merkel et al.,
2002; Duffy et al., 1995). It can be inferred that in a mixture, strong material could
strengthen the “soft” material, whereas “soft” material may also be influenced by the
strong material. It is also noted that the involvement of strong materials (MgO) may
extend the stability of NaCl B1 phase to much higher pressure. Importantly, this study
provides a first systematic study of strength model for MgO-NaCl mixture that has
important implication for the strength and deformation of multi-phase compositions at
deep mantle pressures.
108
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Appendices
Appendix A1 The variations of the d-spacing as a function of 1-3cos2ψ for the pure
NaCl
1.52
B1 222
10.2
1.50
d spacing (Angstrom)
1.48
15.2
1.46
20.1
1.44
26.9
1.42
29.8
1.40
-2.0
-1.5
-1.0
-0.5
2
1-3cos 
0.0
0.5
1.0
FigureA1.1. NaCl B1 222 d-spacings as a function of 1-3cos2ψ at pressure of 10.2-29.8
GPa. Linear fits can be clearly observed. Pressure is labeled with each linear fit.
3.02
B2 100
32.3
d spacing (Angstrom)
3.00
2.98
38.9
2.96
43.7
2.94
2.92
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
2
1-3cos 
FigureA1.2. NaCl B2 100 d-spacings as a function of 1-3cos2ψ at pressure of 32.3-43.7
GPa. Linear fits can be clearly observed. Pressure is labeled with each linear fit.
122
Appendix A 2 The variations of the d-spacing as a function of 1-3cos2ψ for the MgONaCl (1:3) mixture
1.88
d_spacing (angstrom)
1.86
1.84
1.82
1.80
B1_220
1.78
1.76
-2.0
-1.5
-1.0
-0.5
2
1-3cos Ψ
0.0
0.5
1.0
FigureA2.1. NaCl B1 220 d-spacings as a function of 1-3cos2ψ at different pressure show
linear fits.
d_spacing (angstrom)
2.32
2.30
Au 111
2.28
2.26
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
2
1-3cos Ψ
FigureA.2.2. Au 111 d-spacings as a function of 1-3cos2ψ at different pressures show
linear fits.
123
Appendix A 3 The variations of the d-spacing as a function of 1-3cos2ψ for the MgONaCl (1:2) mixture
2.08
d_spacing (angstrom)
MgO (200)
2.06
2.04
2.02
2.00
-2.0
-1.5
-1.0
-0.5
0.0
2
1-3cos ψ
0.5
1.0
FigureA.3.2. MgO 200 d-spacings as a function of 1-3cos2ψ at different pressures show
linear fits.
2.25
d_spacing (angstrom)
2.24
Au 111
2.23
2.22
-2.0
-1.5
-1.0
-0.5
2
1-3cos ψ
0.0
0.5
1.0
FigureA.3.3. Au 111 d-spacings as a function of 1-3cos2ψ at different pressures show
linear fits.
124
Appendix A 4 The variations of the d-spacing as a function of 1-3cos2ψ for the MgONaCl (1:1) mixture
2.33
d_spacing (angstrom)
2.32
Au 111
2.31
2.30
2.29
2.28
2.27
2.26
-2.0
-1.0
2
1-3cos ψ
0.0
1.0
FigureA.4.1. Au 111 d-spacings as a function of 1-3cos2ψ at different pressures show
linear fits.
1.47
d_spacing (angstrom)
MgO 220
1.46
1.45
1.44
1.43
-1.5
-1.0
-0.5
0.0
2
1-3cos ψ
0.5
1.0
FigureA.4.2. MgO 220 d-spacings as a function of 1-3cos2ψ at different pressures show
linear fits.
125
Differential Stress (GPa)
MgO 200
MgO 220 (Kinsland & Bassett, 1977)
small strain
large strain (Meade&Jeanloz, 1988)
Duffy et al.,1995
run 1
run 2 (Merkel et al.,2002)
Uchida et al., 2004
12
Singh et al., 2004
Akhmetov, 2008
Gleason et al.,2011
MgO (1:1), this study
nt_kinsland
nt_Meade 10
Nt_Pure_B1
nt_Pure_B2
nt_Weidner
nt_1to1_B1
nt_1to1_B2 8
nt_1to2_B1
nt_1to2_B2
nt_1to3_B1
nt_1to3_B2
nt_1to4_B1 6
nt_1to4_B2
MgO
MgO in mixture
NaCl in mixture
4
2
0
NaCl
0
10
20
30
Pressure (GPa)
40
50
60
FigureA.4.3. Stresses of MgO, MgO in a mixture, NaCl in a mixture and NaCl as
function of pressures. Orange symbols: differential stress of pure MgO; Blue symbols:
MgO in a mixture. Red symbols: the differential stress of NaCl in a mixture condition.
Green symbols: the differential of stress supported by pure NaCl sample. Solid symbols:
this study. Open symbols: previous published reports.
126
Curriculum Vitae
Name:
Zhongying MI
Post-secondary
Education and
Degrees:
China University of Geosciences
Beijing, China
2001-2005 B.A.
Gemology and Material Science
China University of Geosciences
Beijing, China
2005-2008 M.A.
Geology
The University of Western Ontario
London, Ontario, Canada
2008-2012 Ph.D.
Geophysics
Honors and
Awards:
COMPRES Conference Scholarship, 2011
Geophysics Travel Scholarship, 2010
CLS Conference Scholarship, 2009
Graduate Thesis Award, 2008
The University of Western Ontario International Graduate Student
Scholarship, 2008-2012
Excellent Thesis Award (Master), 2008
Excellent Thesis Award (Undergrad), 2005
Annual Student Scholarship, 2001-2005
Related Work
Experience
Teaching Assistant
The University of Western Ontario
2008-2012
Published Thesis/Abstract:
Mi, Z, Shieh, S.R, Duffy, T.S, Kiefer, B, 2011. Strength of NaCl in
Different Composite Environments, CAMBR Meeting, Pub.
Abstract
Mi, Z, Shieh, S.R, Duffy, T.S, Kiefer, B, 2011. Strength Study on
MgO and NaCl Mixture, COMPRES Annual Meeting, Pub.
Abstract
Mi, Z, Shieh, S.R, Duffy, T.S, Kiefer, B, High Pressure Strength
Study on NaCl, AGU, Abstract MR11C-1891 presented at 2010
Fall Meeting, AGU, San Francisco, Calif., 13-17 Dec